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 1-6 of 6 results.

A134091 Column 1 of triangle A134090.

Original entry on oeis.org

1, 2, 9, 46, 285, 2021, 16023, 139812, 1326111, 13544857, 147880458, 1715413558, 21036674321, 271585117428, 3677831536291, 52081368845176, 769123715337395, 11816582501728389, 188470925178659344, 3114771205613655362
Offset: 0

Views

Author

Paul D. Hanna, Oct 07 2007

Keywords

Comments

Row n of triangle T=A134090 = row n of (I + D*C)^n for n>=0 where C denotes Pascal's triangle, I the identity matrix and D a matrix where D(n+1,n)=1 and zeros elsewhere.

Crossrefs

Cf. A134090; columns: A122455, A134092, A134093; A134094 (row sums); A048993 (S2).

Programs

  • PARI
    a(n)=polcoeff(sum(k=0,n+1,binomial(n+1,k)*x^k/(1-k*x)/prod(i=0,k,1-i*x +x*O(x^n))),n)

Formula

a(n) = [x^n] Sum_{k=0..n+1} C(n+1,k)*x^k/(1-k*x) / [Product_{i=1..k}(1-i*x)].

A134092 Column 2 of triangle A134090.

Original entry on oeis.org

1, 3, 18, 110, 780, 6167, 53494, 504030, 5112090, 55411697, 638154165, 7770348170, 99618149267, 1339889000543, 18848892749144, 276573551651632, 4222814264496510, 66947348027905977, 1099955438013660173
Offset: 0

Views

Author

Paul D. Hanna, Oct 08 2007

Keywords

Comments

Row n of triangle T=A134090 = row n of (I + D*C)^n for n>=0 where C denotes Pascal's triangle, I the identity matrix and D a matrix where D(n+1,n)=1 and zeros elsewhere.

Crossrefs

Cf. A134090; columns: A122455, A134091, A134093; A134094 (row sums); A048993 (S2).

Programs

  • PARI
    {a(n)= polcoeff(sum(k=0,n+2,binomial(n+2,k)*x^k/(1-k*x)^2/prod(i=0,k,1-i*x +x*O(x^n))),n)}

Formula

a(n) = [x^n] Sum_{k=0..n+2} C(n+2,k)*x^k/(1-k*x)^2 / [Product_{i=1..k}(1-i*x)].

A134093 Column 3 of triangle A134090.

Original entry on oeis.org

1, 4, 30, 215, 1729, 15176, 143814, 1462995, 15876410, 182811992, 2223580281, 28458251185, 381943459065, 5359649816728, 78430018675440, 1194057733357517, 18873870914263424, 309154787519651284, 5238840625331179517
Offset: 0

Views

Author

Paul D. Hanna, Oct 08 2007

Keywords

Comments

Row n of triangle T=A134090 = row n of (I + D*C)^n for n>=0 where C denotes Pascal's triangle, I the identity matrix and D a matrix where D(n+1,n)=1 and zeros elsewhere.

Crossrefs

Cf. A134090; columns: A122455, A134091, A134092; A134094 (row sums); A048993 (S2).

Programs

  • PARI
    {a(n)= polcoeff(sum(k=0,n+3,binomial(n+3,k)*x^k/(1-k*x)^3/prod(i=0,k,1-i*x +x*O(x^n))),n)}

Formula

a(n) = [x^n] Sum_{k=0..n+3} C(n+3,k)*x^k/(1-k*x)^3 / [Product_{i=1..k}(1-i*x)].

A122455 a(n) = Sum_{k=0..n} C(n,k)*S2(n,k). Binomial convolution of the Stirling numbers of the 2nd kind. Also sum of the rows of A122454.

Original entry on oeis.org

1, 1, 3, 13, 71, 456, 3337, 27203, 243203, 2357356, 24554426, 272908736, 3218032897, 40065665043, 524575892037, 7197724224361, 103188239447115, 1541604242708064, 23945078236133674, 385890657416861532, 6440420888899573136, 111132957321230896024
Offset: 0

Views

Author

Alford Arnold, Sep 18 2006

Keywords

Comments

A122454(n,k) = A098546(n,k) times A036040(n,k) (two triangles shaped by integer partitions A000041(n)).
Row sums of A098546 give sequence A098545 and row sums of A036040 give sequence A000110 (the Bell numbers)
Equals column zero of triangle A134090: let C equal Pascal's triangle, I the identity matrix and D a matrix where D(n+1,n)=1 and zeros elsewhere; then a(n) = column 0 of row n of (I + D*C)^n (see A134090). - Paul D. Hanna, Oct 07 2007
Number of Green's H-classes in the full transformation semigroup on [n]. Row sums of A090683. - Geoffrey Critzer, Dec 27 2022

Examples

			A098546(n) begins 1 2 1 3 3 1 4 6 6 4 1 ...
A036040(n) begins 1 1 1 1 3 1 1 4 3 6 1 ...
so
A122454(n) begins 1 2 1 3 9 1 4 24 18 24 1 ...
and
the present sequence begins 1 3 13 71 ...
with A000041 entries per row.

References

  • O. Ganyushkin and V. Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, pages 58-62.

Links

Crossrefs

Cf. A000041, A000110, A036040, A098545, A098546, A122454.
Cf. A134090, A048993 (S2).
Cf. A090683.

Programs

  • Magma
    [(&+[Binomial(n,k)*StirlingSecond(n,k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Feb 07 2019
  • Maple
    sortAbrSteg := proc(L1,L2) local i ; if nops(L1) < nops(L2) then RETURN(true) ; elif nops(L2) < nops(L1) then RETURN(false) ; else for i from 1 to nops(L1) do if op(i,L1) < op(i,L2) then RETURN(false) ; fi ; od ; RETURN(true) ; fi ; end: A098546 := proc(n,k) local prts,m ; prts := combinat[partition](n) ; prts := sort(prts, sortAbrSteg) ; if k <= nops(prts) then m := nops(op(k,prts)) ; binomial(n,m) ; else 0 ; fi ; end: M3 := proc(L) local n,k,an,resul; n := add(i,i=L) ; resul := factorial(n) ; for k from 1 to n do an := add(1-min(abs(j-k),1),j=L) ; resul := resul/ (factorial(k))^an /factorial(an) ; od ; end: A036040 := proc(n,k) local prts,m ; prts := combinat[partition](n) ; prts := sort(prts, sortAbrSteg) ; if k <= nops(prts) then M3(op(k,prts)) ; else 0 ; fi ; end: A122454 := proc(n,k) A098546(n,k)*A036040(n,k) ; end: A122455 := proc(n) add(A122454(n,k),k=1..combinat[numbpart](n)) ; end: seq(A122455(n),n=1..18) ; # R. J. Mathar, Jul 17 2007
# Alternatively:
A122455 := n -> add(binomial(n,k)*Stirling2(n,k),k=0..n):
seq(A122455(n),n=0..21); # Peter Luschny, Aug 11 2015
  • Mathematica
    Table[Sum[Binomial[n, k]*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
  • PARI
    a(n)= polcoeff(sum(k=0,n,binomial(n,k)*x^k/prod(i=0,k,1-i*x +x*O(x^n))),n) \\ Paul D. Hanna, Oct 07 2007
  • PARI
    a(n)=sum(k=0,n, binomial(n,k) * stirling(n,k,2) ); /* Joerg Arndt, Jun 16 2012 */
  • Sage
    [sum(binomial(n,k)*stirling_number2(n,k) for k in (0..n)) for n in range(20)] # G. C. Greubel, Feb 07 2019

Formula

a(n) = [x^n] Sum_{k=0..n} C(n,k) * x^k / [Product_{i=0..k} (1 - i*x)]; equivalently, a(n) = Sum_{k=0..n} C(n,k) * S2(n,k), where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind. - Paul D. Hanna, Oct 07 2007

Extensions

More terms from R. J. Mathar, Jul 17 2007
Definition modified by Olivier Gérard, Oct 23 2012
a(0)=1 prepended by Alois P. Heinz, Sep 17 2017

A211210 a(n) = Sum_{k=0..n} binomial(n, k)*|S1(n, k)|.

Original entry on oeis.org

1, 1, 3, 16, 115, 1021, 10696, 128472, 1734447, 25937683, 424852351, 7554471156, 144767131444, 2971727661124, 65013102375404, 1509186299410896, 37032678328740751, 957376811266995031, 25999194631060525009, 739741591417352081464, 22000132609456951524051
Offset: 0

Views

Author

Olivier Gérard, Oct 23 2012

Keywords

Comments

Binomial convolution of the unsigned Stirling numbers of the first kind.
Row sums of triangle A187555.

Crossrefs

Cf. A317274 (signed S1), A187555, A134090, A211211.
Cf. A122455 (second kind), A271702, A134094, A343841 (second kind inverse).

Programs

  • Mathematica
    Table[Sum[Binomial[n, k] Abs[StirlingS1[n, k]], {k, 0, n}], {n, 0, 20}]
  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*abs(stirling(n, k, 1))); \\ Michel Marcus, May 10 2021

A134094 Binomial convolution of the Stirling numbers of the second kind.

Original entry on oeis.org

1, 2, 6, 26, 140, 887, 6405, 51564, 455712, 4370567, 45081476, 496556194, 5806502663, 71734434956, 932447207866, 12707973761320, 181033752071568, 2688530124711819, 41525910256013832, 665674913113633582
Offset: 0

Views

Author

Paul D. Hanna, Oct 08 2007

Keywords

Comments

Row n of triangle T=A134090 = row n of (I + D*C)^n for n>=0 where C denotes Pascal's triangle, I the identity matrix and D a matrix where D(n+1,n)=1 and zeros elsewhere.

Links

Crossrefs

Cf. A134090; columns: A122455, A134091, A134092, A134093; A048993 (S2).
Cf. A000110.

Programs

  • Maple
    f:= proc(n) local k; add(binomial(n+1,k)*combinat:-stirling2(n,k),k=0..n) end proc:
map(f, [$0..30]); # Robert Israel, Oct 16 2019
  • Mathematica
    Table[Sum[Binomial[n + 1, k] StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)*polcoeff((1-k*x)/prod(i=0,k+1,1-i*x+x*O(x^(n))),n-k))}

Formula

a(n) = sum( C(n+1,k)*|S2(n,k)|, k=0..n).
Row sums of triangle A134090.
a(n) = [x^n] Sum_{k=0..n} C(n,k)*x^k*(1-k*x) / [Product_{i=0..k+1}(1-i*x)], equivalently, a(n) = Sum_{k=0..n} C(n,k)*[S2(n,k) - k*S2(n-1,k)], where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind.
a(n) = Sum_{k=0..n} C(n+1,k)*S2(n,k). From Olivier Gérard, Oct 23 2012

Extensions

Definition modified and Mathematica program by Olivier Gérard, Oct 23 2012
Simplified Name and moved formulas into the formula section. - Paul D. Hanna, Oct 23 2013
Showing 1-6 of 6 results.

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