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.

A131222 Exponential Riordan array [1, log((1-x)/(1-2x))].

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 14, 9, 1, 0, 90, 83, 18, 1, 0, 744, 870, 275, 30, 1, 0, 7560, 10474, 4275, 685, 45, 1, 0, 91440, 143892, 70924, 14805, 1435, 63, 1, 0, 1285200, 2233356, 1274196, 324289, 41160, 2674, 84, 1
Offset: 0

Views

Author

Paul Barry, Jun 18 2007

Keywords

Comments

This is also the matrix product of the unsigned Lah numbers and the Stirling cycle numbers. See also A079639 and A079640 for variants based on an (1,1)-offset of the number triangles. - Peter Luschny, Apr 12 2015
The Bell transform of A029767(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016
Essentially the same as A079638. - Peter Bala, Feb 12 2022

Examples

			Number triangle starts:
  1,
  0,   1;
  0,   3,   1;
  0,  14,   9,   1;
  0,  90,  83,  18,  1;
  0, 744, 870, 275, 30,  1;
  ...

Links

Crossrefs

Cf. A000007, A002866, A029767, A079638, A079639, A079640.

Programs

  • Maple
    RioExp := (d,h,n,k) -> coeftayl(d*h^k, x=0,n)*n!/k!:
A131222 := (n,k) -> RioExp(1,log((1-x)/(1-2*x)),n,k):
seq(print(seq(A131222(n,k),k=0..n)),n=0..5); # Peter Luschny, Apr 15 2015
# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n=0,1,n!*(2^(n+1)-1)), 9); # Peter Luschny, Jan 27 2016
  • Mathematica
    BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[If[# == 0, 1, #! (2^(#+1) - 1)]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)
  • Maxima
    T(n,m):=if n=0 and m=0 then 1 else n!*sum((stirling1(k,m)*2^(n-k)*binomial(n-1,k-1))/k!,k,m,n); /* Vladimir Kruchinin, Sep 27 2012 */
  • Sage
    def Lah(n, k):
    if n == k: return 1
    if k<0 or  k>n: return 0
    return (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
matrix(ZZ, 8, Lah) * matrix(ZZ, 8, stirling_number1) # as a square matrix Peter Luschny, Apr 12 2015
# alternatively:
  • Sage
    # uses[bell_matrix from A264428]
bell_matrix(lambda n: A029767(n+1), 10) # Peter Luschny, Jan 18 2016

Formula

Row sums are A002866.
Second column is A029767.
T(n,m) = n! * Sum_{k=m..n} Stirling1(k,m)*2^(n-k)*binomial(n-1,k-1)/k!, n >= m >= 0. - Vladimir Kruchinin, Sep 27 2012

A088815 Expansion of e.g.f. (1-x)^(-1/(1+log(1-x))).

Original entry on oeis.org

1, 1, 4, 24, 190, 1860, 21638, 291158, 4443556, 75779580, 1427272032, 29409572808, 657829667328, 15868725580344, 410543007882408, 11336582934052104, 332736828827893968, 10342443317857993680, 339343476195341474688
Offset: 0

Views

Author

Vladeta Jovovic, Nov 22 2003

Keywords

Links

Crossrefs

Row sums of A079640.
Cf. A000262, A007840, A088819.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[(1-x)^(-1/(1+Log[1-x])), {x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Nov 29 2011 *)
  • PARI
    x='x+O('x^25); Vec(serlaplace((1-x)^(-1/(1+log(1-x))))) \\ G. C. Greubel, Feb 16 2017
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, k!*abs(stirling(j, k, 1)))*binomial(i-1, j-1)*v[i-j+1])); v; \\ Seiichi Manyama, May 23 2022

Formula

a(n) = Sum_{k=0..n} |Stirling1(n, k)|*A000262(k). - Vladeta Jovovic, Nov 26 2003
a(n) ~ n! * exp(n + 2*sqrt(n)/sqrt(exp(1)-1) + 1/(2*(exp(1)-1)) - 1/2) / (2*sqrt(Pi) * (exp(1)-1)^(n+1/4) * n^(3/4)). - Vaclav Kotesovec, May 04 2015
a(0) = 1; a(n) = Sum_{k=1..n} A007840(k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, May 23 2022

A256892 Triangular array read by rows, the matrix product of the unsigned Lah numbers and the Stirling set numbers, T(n,k) for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 13, 9, 1, 0, 73, 79, 18, 1, 0, 501, 755, 265, 30, 1, 0, 4051, 7981, 3840, 665, 45, 1, 0, 37633, 93135, 57631, 13580, 1400, 63, 1, 0, 394353, 1192591, 911582, 274141, 38290, 2618, 84, 1, 0, 4596553, 16645431, 15285313, 5633922, 999831, 92358, 4494, 108, 1
Offset: 0

Views

Author

Peter Luschny, Apr 12 2015

Keywords

Comments

Also the Bell transform of A000262(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

Examples

			Triangle starts:
1;
0,    1;
0,    3,    1;
0,   13,    9,    1;
0,   73,   79,   18,   1;
0,  501,  755,  265,  30,  1;
0, 4051, 7981, 3840, 665, 45, 1;

Crossrefs

See also A088814 and A088729 for variants based on an (1,1)-offset of the number triangles. See A131222 for the product Lah * Stirling-cycle.
A079640 is an unsigned matrix inverse reduced to an (1,1)-offset.
Cf. A000262, A045943, A084357, A088729, A088814.

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
BellMatrix(n -> simplify(hypergeom([-n, -n-1], [], 1)), 9); # Peter Luschny, Jan 29 2016
  • Mathematica
    BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, HypergeometricPFQ[{-n, -n-1}, {}, 1]], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)
  • SageMath
    def Lah(n, k):
    if n == k: return 1
    if k<0 or  k>n: return 0
    return (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
matrix(ZZ, 8, Lah) * matrix(ZZ, 8, stirling_number2)  # as a square matrix

Formula

T(n+1,1) = A000262(n).
T(n+1,n) = A045943(n).
Row sums are A084357.
Showing 1-3 of 3 results.

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