A122455 -id:A122455 - cp's OEIS frontend

A134055 a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(n,k) for n>0, a(0)=1, where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind.

1, 1, 2, 8, 41, 252, 1782, 14121, 123244, 1169832, 11960978, 130742196, 1518514076, 18645970943, 241030821566, 3268214127548, 46338504902485, 685145875623056, 10538790233183702, 168282662416550040, 2784205185437851772, 47646587512911994120

Offset: 0

Paul D. Hanna, Oct 08 2007

nonn

			O.g.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 41*x^4 + 252*x^5 + 1782*x^6 + 14121*x^7 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^2*x^2/(1-2*x)^2*exp(-2*x/(1-2*x))/2! + 3^3*x^3/(1-3*x)^3*exp(-3*x/(1-3*x))/3! + 4^4*x^4/(1-4*x)^4*exp(-4*x/(1-4*x))/4! +...
simplifies to a power series in x with integer coefficients.
Illustrate the definition of the terms by:
a(4) = 1*1 + 3*7 + 3*6 + 1*1 = 41;
a(5) = 1*1 + 4*15 + 6*25 + 4*10 + 1*1 = 252;
a(6) = 1*1 + 5*31 + 10*90 + 10*65 + 5*15 + 1*1 = 1782.

Alois P. Heinz, Table of n, a(n) for n = 0..518

Cf. A037011, A048297, A048993, A122455, A218667, A218670, A245059, A245060.

a:= proc(n) option remember; local b; b:=
      proc(h, m) option remember; `if`(h=0,
        binomial(n-1, m-1), m*b(h-1, m)+b(h-1, m+1) )
      end; b(n, 0)
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Jun 24 2023

Flatten[{1,Table[Sum[Binomial[n-1,k-1] * StirlingS2[n,k],{k,1,n}],{n,1,20}]}] (* Vaclav Kotesovec, Aug 11 2014 *)

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

a(n)=polcoeff(sum(k=0,n+1,(k*x)^k/(1-k*x)^k*exp(-k*x/(1-k*x+x*O(x^n)))/k!),n)
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Nov 04 2012

O.g.f.: Sum_{n>=0} (n*x)^n/(1-n*x)^n * exp(-n*x/(1-n*x)) / n!. - Paul D. Hanna, Nov 04 2012
From Alois P. Heinz, Jun 24 2023: (Start)
a(n) mod 2 = A037011(n) for n >= 1.
a(n) mod 2 = 1 <=> n in { A048297 } or n = 0. (End)

An initial '1' was added and definition changed slightly by Paul D. Hanna, Nov 04 2012

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

Olivier Gérard, Oct 23 2012

nonn

Binomial convolution of the unsigned Stirling numbers of the first kind.

Row sums of triangle A187555.

Cf. A317274 (signed S1), A187555, A134090, A211211.

Cf. A122455 (second kind), A271702, A134094, A343841 (second kind inverse).

Table[Sum[Binomial[n, k] Abs[StirlingS1[n, k]], {k, 0, n}], {n, 0, 20}]

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

Paul D. Hanna, Oct 08 2007

nonn

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.

Robert Israel, Table of n, a(n) for n = 0..517

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

Cf. A000110.

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

Table[Sum[Binomial[n + 1, k] StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]

{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))}

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

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

A134090 Triangle, read by rows, where T(n,k) = [(I + DC)^n](n,k); that is, row n of T = row n of (I + DC)^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.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 13, 9, 3, 1, 71, 46, 18, 4, 1, 456, 285, 110, 30, 5, 1, 3337, 2021, 780, 215, 45, 6, 1, 27203, 16023, 6167, 1729, 371, 63, 7, 1, 243203, 139812, 53494, 15176, 3346, 588, 84, 8, 1, 2357356, 1326111, 504030, 143814, 32376, 5886, 876, 108, 9, 1

Offset: 0

Paul D. Hanna, Oct 07 2007

Column 0 equals A122455 if we define A122455(0)=1.

			Triangle T begins:
1;
1, 1;
3, 2, 1;
13, 9, 3, 1;
71, 46, 18, 4, 1;
456, 285, 110, 30, 5, 1;
3337, 2021, 780, 215, 45, 6, 1;
27203, 16023, 6167, 1729, 371, 63, 7, 1;
243203, 139812, 53494, 15176, 3346, 588, 84, 8, 1;
2357356, 1326111, 504030, 143814, 32376, 5886, 876, 108, 9, 1; ...
Let P denote the matrix equal to Pascal's triangle shift down 1 row:
P(n,k) = C(n+1,k) for n>k>=0, with P(n,n)=1 for n>=0.
Illustrate row n of T = row n of P^n as follows.
Matrix P = I + D*C begins:
1;
1, 1;
1, 1, 1;
1, 2, 1, 1;
1, 3, 3, 1, 1;
1, 4, 6, 4, 1, 1; ...
Matrix cube P^3 begins:
1;
3, 1;
6, 3, 1;
13, 9, 3, 1; <== row 3 of P^3 = row 3 of T
30, 25, 12, 3, 1;
73, 72, 40, 15, 3, 1; ...
Matrix 4th power P^4 begins:
1;
4, 1;
10, 4, 1;
26, 14, 4, 1;
71, 46, 18, 4, 1; <== row 4 of P^4 = row 4 of T
204, 155, 70, 22, 4, 1; ...
Matrix 5th power P^5 begins:
1;
5, 1;
15, 5, 1;
45, 20, 5, 1;
140, 75, 25, 5, 1;
456, 285, 110, 30, 5, 1; <== row 5 of P^5 = row 5 of T.

Cf. columns: A134091, A134092, A134093; A134094 (row sums).

\\ As generated by the g.f.
{T(n,k)=polcoeff(sum(j=0,n,binomial(n,j)*x^j/(1-j*x)^k/prod(i=0,j,1-i*x+x*O(x^(n-k)))),n-k)}

\\ As generated by matrix power: row n of T equals row n of P^n
{T(n,k)=local(P=matrix(n+1,n+1,r,c,if(r==c,1,if(r>c,binomial(r-2,c-1)))));(P^n)[n+1,k+1]}

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

A317274 a(n) = Sum_{k=0..n} binomial(n,k)*Stirling1(n,k).

Original entry on oeis.org

1, 1, -1, -2, 19, -79, 76, 2640, -36945, 329371, -1861949, -4438774, 355714228, -7292531180, 109844527612, -1277006731104, 8181112825231, 124379387459175, -6806984421310187, 191750786928500050, -4289244423048443149, 80163499107525756105, -1146313133241947091420, 5494990440819210736560

Offset: 0

Ilya Gutkovskiy, Jul 25 2018

sign

Binomial convolution of the signed Stirling numbers of the first kind.

Cf. A008275, A048994, A122455, A211210, A211211.

Table[Sum[Binomial[n, k] StirlingS1[n, k], {k, 0, n}], {n, 0, 23}]

a(n) = sum(k=0, n, binomial(n, k)*stirling(n, k, 1)); \\ Michel Marcus, Aug 07 2019

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

Paul D. Hanna, Oct 07 2007

nonn

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.

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

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)

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

Paul D. Hanna, Oct 08 2007

nonn

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.

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

{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)}

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

Paul D. Hanna, Oct 08 2007

nonn

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.

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

{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)}

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)].

A090683 Triangle read by rows, defined by T(n,k) = C(n,k)*S2(n,k), 0 <= k <= n, where C(n,k) are the binomial coefficients and S2(n,k) are the Stirling numbers of the second kind.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 9, 1, 0, 4, 42, 24, 1, 0, 5, 150, 250, 50, 1, 0, 6, 465, 1800, 975, 90, 1, 0, 7, 1323, 10535, 12250, 2940, 147, 1, 0, 8, 3556, 54096, 119070, 58800, 7448, 224, 1, 0, 9, 9180, 254100, 979020, 875826, 222264, 16632, 324, 1, 0, 10, 22995, 1119600, 7162050, 10716300, 4793670, 705600, 33750, 450, 1

Offset: 0

Philippe Deléham, Dec 18 2003

T(n,k) is the number of Green's H-classes contained in the D-class of rank k in the full transformation semigroup on [n]. - Geoffrey Critzer, Dec 27 2022

			Triangle begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  9,  1;
  0,  4, 42, 24,  1;
  ...

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

Vincenzo Librandi, Table of n, a(n) for n = 0..230
Wikipedia, Green's relations
Wikipedia, Transformation semigroup

Cf. A007318, A048993, A090657.

Row sum sequence is A122455.

Flatten[Table[Table[Binomial[n, k] StirlingS2[n, k], {k, 0, n}], {n, 0, 10}], 1]

create_list(binomial(n,k)*stirling2(n,k),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */

T(n, k) = binomial(n,k)*Stirling2(n,k).
T(n, k) = A007318(n, k)*A048993(n, k).
T(n, k) = A090657(n, k)/k!.

Edited by Olivier Gérard, Oct 23 2012

A047799 a(n) = Sum_{k=0..n} C(n,k)*Stirling1(n,k)^2.

Original entry on oeis.org

1, 1, 3, 40, 1015, 40631, 2334766, 180836664, 18067408311, 2254675244287, 342877692847261, 62311687363814736, 13318714515734069806, 3304254169559017642774, 940912768920331123369272, 304601441677789509306775856

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..245

Cf. A008275, A047798, A122455.

List([0..20], n-> Sum([0..n], k-> Binomial(n,k)*Stirling1(n,k)^2 )) # G. C. Greubel, Aug 07 2019

[(&+[Binomial(n,k)*StirlingFirst(n,k)^2: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 07 2019

seq(add(binomial(n,k)*stirling1(n, k)^2, k = 0..n), n = 0..20); # G. C. Greubel, Aug 07 2019

Table[Sum[Binomial[n, k]*StirlingS1[n, k]^2, {k, 0, n}], {n,0,20}] (* G. C. Greubel, Aug 07 2019 *)

{a(n) = sum(k=0,n, binomial(n,k)*stirling(n,k,1)^2)};
vector(20, n, n--; a(n)) \\ G. C. Greubel, Aug 07 2019

[sum(binomial(n,k)*stirling_number1(n,k)^2 for k in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 07 2019

cp's OEIS Frontend

A134055 a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(n,k) for n>0, a(0)=1, where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind.

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

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

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A134090 Triangle, read by rows, where T(n,k) = [(I + D*C)^n](n,k); that is, row n of T = 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.

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A317274 a(n) = Sum_{k=0..n} binomial(n,k)*Stirling1(n,k).

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A134091 Column 1 of triangle A134090.

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A134092 Column 2 of triangle A134090.

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A134093 Column 3 of triangle A134090.

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A134090 Triangle, read by rows, where T(n,k) = [(I + DC)^n](n,k); that is, row n of T = row n of (I + DC)^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.