cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A136656 Coefficients for rewriting generalized falling factorials into ordinary falling factorials.

Original entry on oeis.org

1, 0, -2, 0, 6, 4, 0, -24, -36, -8, 0, 120, 300, 144, 16, 0, -720, -2640, -2040, -480, -32, 0, 5040, 25200, 27720, 10320, 1440, 64, 0, -40320, -262080, -383040, -199920, -43680, -4032, -128, 0, 362880, 2963520, 5503680, 3764880, 1142400, 163968, 10752, 256, 0, -3628800, -36288000, -82978560
Offset: 0

Views

Author

Wolfdieter Lang, Feb 22 2008, Sep 09 2008

Keywords

Comments

Generalization of (signed) Lah number triangle A008297 (amended with a trivial row n=0 and a column k=0 in order to have a Sheffer triangle structure of the Jabotinsky type).
product(s*t-j,j=0..n-1) := fallfac(s*t,n) (falling factorial with n factors) is called generalized factorial of t of order n and scale parameter s in the Charalambides reference p. 301 ch. 8.4.
The s-family of triangles L(s;n,k) (in the Charalambides reference called C(n,k;-s)) is defined for integer s by fallfac(-s*t,n) = ((-1)^n)*risefac(s*t,n) = sum(L(s;n,k)*fallfac(t,k),k=0..n), n>=0. risefac(x,n):=product(x+j,j=0..n-1) for the rising factorials.
For positive s the signless triangles |L(s;n,k)| = L(s;n,k)*(-1)^n satisfies risefac(s*t,n) = sum(|L(s;n,k)|*fallfac(t,k),k=0..n), n>=0.
For negative s see the combinatorial interpretation given in the Charalambides reference, Example 8.8, p. 313: Coupon collector's problem.
|T(n,k)| = B_{n,k}((j+2)!; j>=0) where B_{n,k} are the partial Bell polynomials. - Peter Luschny, May 11 2015

Examples

			Triangle starts:
[1]
[0,   -2]
[0,    6,    4]
[0,  -24,  -36,   -8]
[0,  120,  300,  144,  16]
...
Recurrence: a(4,2) = -7*a(3,2)-2*a(3,1) = -7*(-36) -2*(-24) = 300.
a(4,2)=300 as sum over the M3 numbers A036040 for the 2 parts partitions of 4: 4*fallfac(-2,1)^1*fallfac(-2,3)^1 + 3*fallfac(-2,2)^2 = 4*(-2)*(-24)+3*6^2 = 300.
Row n=3: [0,-24,-36,-8] for the coefficients in rewriting fallfac(-2*t,3)=((-1)^3)*risefac(2*t,3) = ((-1)^3)*(2*t)*(2*t+1)*(2*t+2) = 0*1 -24*t -36*t*(t-1) -8*t*(t-1)*(t-2).

References

  • Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, ch. 8.4 p. 301 ff. Table 8.3 (with row n=0 and column k=0 and s=-2).

Links

Crossrefs

Column sequences (unsigned) 2*A001710, 4*A136659, 8*A136660, 16*A136661 for k=1..4.
Cf. A136657 without row n=0 and column k=0, divided by 2.

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
BellMatrix(n -> (-1)^(n+1)*(n+2)!, 9); # Peter Luschny, Jan 27 2016
  • Mathematica
    fallfac[n_, k_] := Pochhammer[n - k + 1, k]; a[n_, k_] := Sum[(-1)^(k - r)*Binomial[k, r]*fallfac[-2*r, n], {r, 0, k}]/k!; Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 09 2013 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[(-1)^(# + 1)*(# + 2)!&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (*Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
  • Sage
    @CachedFunction
def T(n, k):  # unsigned case
    if k == 0: return 1 if n == 0 else 0
    return sum(T(n-j-1,k-1)*(j+1)*(j+2)*gamma(n)/gamma(n-j) for j in (0..n-k))
for n in range(7): [T(n,k) for k in (0..n)] # Peter Luschny, Mar 31 2015

Formula

Recurrence: a(n,k) = 0 if n
E.g.f. column k: (1/(1+x)^2 - 1)^k/k!, k>=0. From the Charalambides reference Theorem 8.14, p. 305 for s=-2.(hence a Sheffer triangle of Jabotinsky type).
a(n,k) = sum(((-1)^(k-r))*binomial(k,r)*fallfac(-2*r,n),r=0..k)/k!, n>=k>=0. From the Charalambides reference Theorem 8.15, p. 306 for s=-2.
a(n,k) = sum(S1(n,r)*S2(r,k)*(-2)^r,r=k..n) with the Stirling numbers S1(n,r)= A048993(n,r) and S2(r,k)= A048993(r,k). From the Charalambides reference Theorem 8.13, p.304 for s=-2.
a(n,k) = sum(M_3(n,k,q)*product(fallfac(-2,j)^e(n,m,q,j),j=1..n),q=1..p(n,k)) if n>=k>=1, else 0. Here p(n,k)=A008284(n,m), the number of k parts partitions of n, the M_3 partition numbers are given in A036040 and e(n,m,q,j) is the exponent of j in the q-th k parts partition of n. Rewritten eq. (8.50), Theorem 8.16, p. 307, from the Charalambides reference for s=-2.
Recurrence for the unsigned case: a(n,k) = Sum_{j=0..n-k} a(n-j-1,k-1)*C(n-1,j)*(j+2)! if k<>0 else k^n. - Peter Luschny, Mar 31 2015

A136660 Unsigned fourth column (k=3) of triangle A136656 divided by 8.

Original entry on oeis.org

1, 18, 255, 3465, 47880, 687960, 10372320, 164656800, 2754259200, 48518870400, 899026128000, 17495593315200, 356995102464000, 7625049239808000, 170196434343936000, 3963602854987776000, 96160451873181696000
Offset: 0

Views

Author

Wolfdieter Lang, Feb 22 2008

Keywords

Comments

Also unsigned third column of triangle A136657 divided by 4.

References

  • Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, Table 8.3, p. 311, with s=-2, k=3 column/8.

Crossrefs

Cf. A136656, A136659 (column k=2), A136661 (column k=4).

Formula

a(n) = |A136656(n+3,3)|/8, n>=0.
E.g.f.: (2+18*x+3*x^2-12*x^3+3*x^4)/(2*(1-x)^9) (derived from the one given for the column k=3 under A136656).

A226167 Array read by antidiagonals: a(i,j) is the number of ways of labeling a tableau of shape (i,1^j) with the integers 1, 2, ... i+j-2 (each label being used once) such that the first row is decreasing, and the first column has m-1 labels.

Original entry on oeis.org

1, 3, 1, 12, 5, 1, 60, 27, 7, 1, 360, 168, 48, 9, 1, 2520, 1200, 360, 75, 11, 1, 20160, 9720, 3000, 660, 108, 13, 1, 181440, 88200, 27720, 6300, 1092, 147, 15, 1, 1814400, 887040, 282240, 65520, 11760, 1680, 192, 17, 1, 19958400, 9797760, 3144960, 740880, 136080, 20160, 2448, 243, 19, 1
Offset: 1

Views

Author

John M. Campbell, May 29 2013

Keywords

Comments

For an arbitrary composition c, let F_c^p denote the linear transformation of NSym that is adjoint to multiplication by the fundamental quasi-symmetric function indexed by c. Then a(i,j) equals the coefficient of H_(1,1) in (F_(1)^p)^(i+j-2)(H_(i,1^j)) (see below SAGE program, and Corollary 2.7 in the below link).
Let M(n) = [a(i,j)]_{n x n}. Then det(M(n))=A000178(n)=the n-th superfactorial.
Let p_n(x) denote the polynomial such that a(x,n)=p_n(x). Then the coefficient of x in p_n(x) is |A009575(n)|. For example, p_4(x)=4x^3+18x^2+26x+12, and the coefficient of x in p_4(x) is |A009575(4)|=26.
First row is A001710. Second row is A138772. Fourth row is A136659.

Examples

			There are a(3,2) = 7 ways of labeling the tableau of shape (3,1,1) with 1, 2 and 3 (with each label being used once) such that the first row is decreasing and the first column has 1 label:
1    2    3    X    X    X    X
X    X    X    1    2    3    X
X32  X31  X21  X32  X31  X21  321
The matrix [a(i,j)]_(6 x 6) is given below:
[1  3  12   60   360   2520]
[1  5  27  168  1200   9720]
[1  7  48  360  3000  27720]
[1  9  75  660  6300  65520]
[1 11 108 1092 11760 136080]
[1 13 147 1680 20160 257040]

Links

Crossrefs

Cf. A000178, A009575.
Main diagonal gives: A023999. - Alois P. Heinz, Jan 21 2014

Programs

  • Maple
    a:= (i, j)-> (i+j-2)!/i!*(2*i+j-1)*j/2:
seq(seq(a(i, 1+d-i), i=1..d), d=1..12);  # Alois P. Heinz, Jan 21 2014
  • Mathematica
    a[n_,k_]:=(n+k-2)!/n!*(2*n+k-1)*k/2 ;
Print[Array[a[#1,#2]&,{50,50}]//MatrixForm]
(* A program which gives a list of tableaux *)
a[i_, j_] :=  Module[{f, list1, el, emptylist, n},
  f[q_] := StringReplace[StringReplace[StringReplace[    StringReplace[ToString[q], ToString[i + j - 1] -> "X"], ", " -> ""], "{" -> ""], "}" -> ""]; list1 = Permutations[Join[Table[q, {q, 1, i + j - 2}], {i + j - 1, i + j - 1}]]; el[q_] := First[Take[list1, {q, q}]]; emptylist = {}; n = 1; While[n < 1 + Length[list1], If[Take[el[n], {j + 1, i + j}] == Sort[Take[el[n], {j + 1, i + j}], Greater] && Count[Take[el[n], {1, j + 1}], i + j - 1] == 2, emptylist = Append[emptylist, f[el[n]]], Null]; n++]; Print[emptylist]]
  • Sage
    NSym = NonCommutativeSymmetricFunctions(QQ) ;
QSym = QuasiSymmetricFunctions(QQ) ;
F = QSym.Fundamental() ;
H = NSym.complete() ;
def a(n, m):
     expr = H([n]+[1 for q in range(m)]) ;
     w=1 ;
     while w

Formula

a(i,j) = (i+j-2)!/i!*(2*i+j-1)*j/2.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.