A134093 -id:A134093 - cp's OEIS frontend

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

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)].

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)].

cp's OEIS Frontend

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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A134091 Column 1 of triangle A134090.

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A134092 Column 2 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.