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  1. "File COMBINAT.MTH, copyright (c) 1990-2001 by Texas Instruments Incorporated"
  2.  
  3. CATALAN(n):=COMB(2*n,n)/(n+1)
  4.  
  5. STIRLING(n,k,k_:=0,u_,v_:=[1]):=PROG(IF(n*k<0,RETURN(0)),IF(n>0,PROG(IF(n<=k,R~
  6. ETURN(MAX(0,n-k+1))),u_:=VECTOR((-1)^(k-j_)*j_^n,j_,0,k),n:=k,LOOP(IF(k=0,RETU~
  7. RN(u_*v_/n!)),v_:=ADJOIN(0,v_)+APPEND(v_,[0]),k:-1))),n:=-n,k:=-k,IF(n>=k,RETU~
  8. RN(MAX(k-n+1,0))),IF(n=0,RETURN(0)),IF(n=1,RETURN((k-1)!)),IF(n=k-1,RETURN(COM~
  9. B(k,2))),IF(n>k/2,PROG(LOOP(IF(k_=k-n,exit),v_:=k_*APPEND(v_,[0])+ADJOIN(0,v_)~
  10. ,k_:+1),LOOP(IF(k_=n,exit),v_:=REST(k_*APPEND(v_,[0])+ADJOIN(0,v_)),k_:+1),LOO~
  11. P(IF(k_=k,RETURN(FIRST(v_))),v_:=DELETE(REST(k_*APPEND(v_,[0])+ADJOIN(0,v_)),-~
  12. 1),k_:+1))),LOOP(IF(k_=n,exit),v_:=k_*ADJOIN(0,v_)+APPEND(v_,[0]),k_:+1),LOOP(~
  13. IF(k_=k-n,exit),v_:=REST(k_*ADJOIN(0,v_)+APPEND(v_,[0])),k_:+1),LOOP(IF(k_=k,R~
  14. ETURN(FIRST(v_))),v_:=DELETE(REST(k_*ADJOIN(0,v_)+APPEND(v_,[0])),-1),k_:+1))
  15.  
  16. STIRLING1(n,k):=STIRLING(-k,-n)
  17.  
  18. STIRLING2(n,k):=STIRLING(n,k)
  19.  
  20. STIRLING_SUBSET(n,k):=STIRLING(n,k)
  21.  
  22. STIRLING_CYCLE(n,k):=STIRLING(-k,-n)
  23.  
  24. BERNOULLI(n):=IF(n=0,1,IF(n=1,-1/2,-n*ZETA(1-n)))
  25.  
  26. BERNOULLI_POLY(n,x):=IF(n=1,x-1/2,n*LIM(SUM(k^(n-1),k,1,x_-1),x_,x)+BERNOULLI(~
  27. n))
  28.  
  29. EULER_POLY(n,x):=2/(n+1)*(BERNOULLI_POLY(n+1,x)-2^(n+1)*BERNOULLI_POLY(n+1,x/2~
  30. ))
  31.  
  32. EULER(n):=IF(MOD(n,2)=1,0,2^n*EULER_POLY(n,1/2))
  33.  
  34. PARTS_AUX(n,m):=IF(n<2*m,1,1+SUM(PARTS_AUX(n-k_,k_),k_,m,FLOOR(n,2)))
  35.  
  36. PARTS(n):=IF(n<0,0,PARTS_AUX(n,1))
  37.  
  38. DISTINCT_PARTS_AUX(n,m):=IF(n<2*m,1,1+SUM(DISTINCT_PARTS_AUX(n-k_,k_+1),k_,m,F~
  39. LOOR(n,2)))
  40.  
  41. DISTINCT_PARTS(n):=IF(n<1,0,DISTINCT_PARTS_AUX(n-1,1))
  42.  
  43. PARTS(n):=IF(n<2,1,FLOOR(APPROX(SUM(1/pi*SQRT(k_/2)*SUM(IF(GCD(h_,k_)=1,COS(pi~
  44. *(SUM((i_/k_-1/2)*(MOD(i_*h_/k_)-1/2),i_,1,k_-1)-2*n*h_/k_)),0),h_,1,k_)*(-2*S~
  45. QRT(6)*#e^(-pi*SQRT(24*n-1)/(6*k_))*(#e^(pi*SQRT(24*n-1)/(3*k_))*(6*k_-pi*SQRT~
  46. (24*n-1))-6*k_-pi*SQRT(24*n-1))/(k_*(24*n-1)^(3/2))),k_,1,SQRT(n)/LOG(n,11)),L~
  47. OG(1/(4*n*SQRT(3))*EXP(pi*SQRT(2*n/3)),10)+5)+0.5))
  48.  
  49. PARTS_LIST(n):=REVERSE((ITERATE(IF(n<DIM(p_),[p_,v_,m_],[APPEND([p_*v_],p_),AP~
  50. PEND(v_,[IF((3*m_^2-m_)/2=DIM(p_)+1 OR (3*m_^2+m_)/2=DIM(p_)+1,(-1)^(m_+1),0)]~
  51. ),IF((3*m_^2+m_)/2=DIM(p_)+1,m_+1,m_)]),[p_,v_,m_],[[1],[1],1],n)) SUB 1)
  52.  
  53. DISTINCT_PARTS(n):=SUM(IF(8*n-16*i_+1=FLOOR(SQRT(8*n-16*i_+1))^2,PARTS(i_),0),~
  54. i_,0,n/2)
  55.  
  56. BELL(n,b:=[1],bc:=[1]):=LOOP(IF(n=0,RETURN(b SUB 1)),b:=ADJOIN(b*bc,b),bc:=ADJ~
  57. OIN(0,bc)+INSERT(0,bc,0),n:-1)
  58.  
  59. RECURRENCE(u,v,v0,m):=REVERSE(ITERATE(APPEND([u],v),v,v0,m-DIM(v0)))
  60.  
  61. "The nth p-sided dth-dimensional polygonal pyramid number, p>=2:"
  62.  
  63. POLYGONAL_PYRAMID(n,p,d):=COMB(n+d-2,d-1)*(n*(p-2)-p+d+2)/d
  64.  
  65. "The nth p-sided polygonal number, p>=2:"
  66.  
  67. POLYGONAL(n,p):=POLYGONAL_PYRAMID(n,p,2)
  68.  
  69. TRIANGULAR(n):=POLYGONAL(n,3)
  70.  
  71. TETRAHEDRAL(n):=POLYGONAL_PYRAMID(n,3,3)
  72.  
  73. PENTATOPE(n):=POLYGONAL_PYRAMID(n,3,4)
  74.  
  75. "The nth p-sided dth-dimensional centered pyramid number, p>=2:"
  76.  
  77. CENTERED_PYRAMID(n,p,d):=COMB(n+d-3,d-2)+p*COMB(n+d-2,d)
  78.  
  79. "The nth p-sided centered number, p>=2:"
  80.  
  81. CENTERED(n,p):=CENTERED_PYRAMID(n,p,2)
  82.  
  83. HEX(n):=CENTERED(n,6)
  84.  
  85. STAR(n):=CENTERED(n,12)
  86.  
  87. OCTAHEDRAL(n):=CENTERED_PYRAMID(n,4,3)
  88.  
  89. CENTERED_CUBE(n,d):=n^d+(n-1)^d
  90.  
  91. CENTERED_HEX(n,d):=n^(d+1)-(n-1)^(d+1)
  92.  
  93. RHOMBIC_DODECAHEDRAL(n):=CENTERED_HEX(n,3)
  94.