A355605 -id:A355605 - cp's OEIS frontend

A355603 Expansion of e.g.f. (1 + x)^(x^4/24).

1, 0, 0, 0, 0, 5, -15, 70, -420, 3024, -22050, 202950, -2113650, 24324300, -305645340, 4174483950, -61253992800, 961049212200, -16054949350440, 284505099278400, -5329752594075000, 105239780964864000, -2184466455408699000, 47550052231211237400

Offset: 0

Seiichi Manyama, Jul 09 2022

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Seiichi Manyama, Table of n, a(n) for n = 0..451

Cf. A007113, A355605.

my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^(x^4/24)))

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4/24*log(1+x))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!/24*sum(j=5, i, (-1)^j*j/(j-4)*v[i-j+1]/(i-j)!)); v;

a(n) = n!*sum(k=0, n\5, stirling(n-4*k, k, 1)/(24^k*(n-4*k)!));

a(0) = 1; a(n) = -(n-1)!/24 * Sum_{k=5..n} (-1)^k * k/(k-4) * a(n-k)/(n-k)!.

a(n) = n! * Sum_{k=0..floor(n/5)} Stirling1(n-4*k,k)/(24^k * (n-4*k)!).

A355619 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 + x)^(x^k/k!).

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, -3, 0, 1, 0, 0, 3, 20, 0, 1, 0, 0, 0, -6, -90, 0, 1, 0, 0, 0, 4, 20, 594, 0, 1, 0, 0, 0, 0, -10, 0, -4200, 0, 1, 0, 0, 0, 0, 5, 40, -126, 34544, 0, 1, 0, 0, 0, 0, 0, -15, -210, 1260, -316008, 0, 1, 0, 0, 0, 0, 0, 6, 70, 1904, -4320, 3207240, 0

Offset: 0

Seiichi Manyama, Jul 10 2022

			Square array begins:
  1,   1,  1,   1,   1, 1, 1, ...
  1,   0,  0,   0,   0, 0, 0, ...
  0,   2,  0,   0,   0, 0, 0, ...
  0,  -3,  3,   0,   0, 0, 0, ...
  0,  20, -6,   4,   0, 0, 0, ...
  0, -90, 20, -10,   5, 0, 0, ...
  0, 594,  0,  40, -15, 6, 0, ...

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

Columns k=1..4 give A007113, A355605, (-1)^n * A351493(n), A355603.

Cf. A355607, A355610.

T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 1)/(k!^j*(n-k*j)!));

T(0,k) = 1 and T(n,k) = -(n-1)!/k! * Sum_{j=k+1..n} (-1)^(j-k) * j/(j-k) * T(n-j,k)/(n-j)! for n > 0.

T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling1(n-k*j,j)/(k!^j * (n-k*j)!).

cp's OEIS Frontend

A355603 Expansion of e.g.f. (1 + x)^(x^4/24).

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A355619 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 + x)^(x^k/k!).

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