A202825 -id:A202825 - cp's OEIS frontend

A294042 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp((1+x)^k - 1).

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 6, 1, 0, 1, 4, 15, 20, 1, 0, 1, 5, 28, 87, 76, 1, 0, 1, 6, 45, 232, 585, 312, 1, 0, 1, 7, 66, 485, 2248, 4383, 1384, 1, 0, 1, 8, 91, 876, 6145, 24544, 35919, 6512, 1, 0, 1, 9, 120, 1435, 13716, 88245, 295456, 318195, 32400, 1, 0

Offset: 0

Seiichi Manyama, Oct 22 2017

			Square array A(n,k) begins:
   1, 1,   1,    1,     1,     1, ...
   0, 1,   2,    3,     4,     5, ...
   0, 1,   6,   15,    28,    45, ...
   0, 1,  20,   87,   232,   485, ...
   0, 1,  76,  585,  2248,  6145, ...
   0, 1, 312, 4383, 24544, 88245, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..5 give A000007, A000012, A000898, A192989, A202824, A202825.
Rows n=0..2 give A000012, A001477, A000384.
Main diagonal gives A294045.
Cf. A000110, A293991.

A(0,k) = 1 and A(n,k) = k * (n-1)! * Sum_{j=1..min(k,n)} binomial(k-1,j-1) * A(n-j,k)/(n-j)! for n > 0.

A(n,k) = Sum_{j=0..n} k^j * Stirling1(n,j) * Bell(j). - Seiichi Manyama, Jan 31 2024

A202824 Expansion of e.g.f.: exp( (1+x)^4 - 1 ).

Original entry on oeis.org

1, 4, 28, 232, 2248, 24544, 295456, 3869632, 54555328, 821239552, 13115934976, 221076780544, 3915685846528, 72609585620992, 1405168845395968, 28302270409560064, 591874919018500096, 12824294700196052992, 287350628454224478208, 6647086535396002004992

Offset: 0

Paul D. Hanna, Dec 25 2011

nonn

			E.g.f.: A(x) = 1 + 4*x + 28*x^2/2! + 232*x^3/3! + 2248*x^4/4! +...
where A(x) = exp(4*x + 6*x^2 + 4*x^3 + x^4).

Seiichi Manyama, Table of n, a(n) for n = 0..501

Cf. A000110, A000898, A008275, A192989, A202825.

List([0..20], n-> Sum([0..n], k-> (-1)^(n-k)*4^k*Bell(k)* Stirling1(n,k) )); # G. C. Greubel, Jul 25 2019

[(&+[4^k*Bell(k)*StirlingFirst(n,k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jul 25 2019

CoefficientList[Series[Exp[(1+x)^4-1], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, May 23 2013 *)
Table[Sum[ (-1)^(n - k) Abs[StirlingS1[n, k]] 4^k BellB[k], {k, 0, n}], {n, 0, 100}] (* Emanuele Munarini, Aug 31 2017 *)

a(n) := sum((-1)^(n-k)*abs(stirling1(n,k))*4^k*belln(k),k,0,n);
makelist(a(n),n,0,12); /* Emanuele Munarini, Aug 31 2017 */

{a(n)=n!*polcoeff(exp((1+x +x*O(x^n))^4-1),n)}

{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k) * 4^k)}

[sum((-1)^(n-k)*4^k*bell_number(k)*stirling_number1(n,k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Jul 25 2019

a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(k) * 4^k.
a(n) ~ n^(3*n/4)*2^(n/2-1)*exp(-3/4+5/16*sqrt(2)*n^(1/4)+sqrt(2)*n^(3/4)-3/4*n+3/4*sqrt(n)). - Vaclav Kotesovec, May 23 2013
a(n+4) - 4*a(n+3) - 12*(n+3)*a(n+2) - 12*(n+2)*(n+3)*a(n+1) - 4*(n+1)*(n+2)*(n+3)*a(n) = 0. - Emanuele Munarini, Aug 31 2017

A369753 Expansion of e.g.f. exp(1 - (1+x)^5).

Original entry on oeis.org

1, -5, 5, 115, -95, -8245, -21275, 896275, 8801825, -95466725, -2703832475, -3522650125, 717727962625, 9961465952875, -118944021914875, -5634631318806125, -37511809003469375, 2020875725751906875, 55489065505990733125, -65182838564153418125

Offset: 0

Seiichi Manyama, Jan 30 2024

sign

Column k=5 of A369738.

Cf. A000587, A202825.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(1-(1+x)^5)))

a(0) = 1; a(n) = -5 * (n-1)! * Sum_{k=1..min(5,n)} binomial(4,k-1) * a(n-k)/(n-k)!.

a(n) = Sum_{k=0..n} 5^k * Stirling1(n,k) * A000587(k).

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A294042 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp((1+x)^k - 1).

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A202824 Expansion of e.g.f.: exp( (1+x)^4 - 1 ).

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A369753 Expansion of e.g.f. exp(1 - (1+x)^5).

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