A052841 -id:A052841 - cp's OEIS frontend

A372245 Triangular array T(n,k) read by rows: column k is the expansion of e.g.f: exp(-2x)(exp(x)-1)^k/(2-exp(x)).

1, -1, 1, 3, -1, 2, -1, 7, 0, 6, 27, 11, 26, 12, 24, 119, 151, 120, 150, 120, 120, 1203, 1139, 1202, 1140, 1200, 1080, 720, 11759, 11887, 11760, 11886, 11760, 11760, 10080, 5040, 136587, 136331, 136586, 136332, 136584, 136080, 131040, 100800, 40320, 1771559, 1772071, 1771560, 1772070

Offset: 0

Thomas Scheuerle, Apr 26 2024

			Triangle T(n, k) starts:
[0]  1;
[1] -1,      1;
[2]  3,     -1,      2;
[3] -1,      7,      0,      6;
[4]  27,     11,     26,     12,     24;
[5]  119,    151,    120,    150,    120,    120;
[6]  1203,   1139,   1202,   1140,   1200,   1080,   720;
[7]  11759,  11887,  11760,  11886,  11760,  11760,  10080,  5040;
[8]  136587, 136331, 136586, 136332, 136584, 136080, 131040, 100800, 40320;

Cf. A000670, A052841, A084416, A341091, A344037.

T(n, k) = sum(m=0, n, ((-1)^((k > 0)+m+n)*binomial(n, m)*(2^(n-m)-(k > 0))*sum(h=max(k-1,0), m, h!*stirling(m, h, 2))))

T(n, k) = Sum_{m=0..n} ((-1)^(1+m+n)*binomial(k, n)*(2^(k - n) - 1)*A084416(m, k - 1)), for k > 0.
T(n, 0) = A344037(n).
T(n, 1) = A052841(n) - A344037(n).
T(n, 2) = A344037(n) - 2*A052841(n) + A000670(n).

A373940 Expansion of e.g.f. 1/(1 - (exp(x) - 1)^5).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, 1800, 16800, 126000, 834120, 8731800, 229191600, 6352632000, 143603580120, 2736395461800, 47283190718400, 860150574738000, 20236134851478120, 614854122909391800, 19930647062659477200, 615406024970593164000, 17883373100352330768120

Offset: 0

Seiichi Manyama, Aug 27 2024

nonn

Cf. A000670, A052841, A353774, A353775.

Cf. A353200.

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i, j)*stirling(j, 5, 2)*v[i-j+1])); v;

a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2));

G.f.: Sum_{k>=0} (5*k)! * x^(5*k)/Product_{j=1..5*k} (1 - j * x).
a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n,k) * Stirling2(k,5) * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k).
a(n) ~ n! / (10 * log(2)^(n+1)). - Vaclav Kotesovec, Aug 27 2024

cp's OEIS Frontend

A372245 Triangular array T(n,k) read by rows: column k is the expansion of e.g.f: exp(-2x)(exp(x)-1)^k/(2-exp(x)).

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A373940 Expansion of e.g.f. 1/(1 - (exp(x) - 1)^5).

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cp's OEIS Frontend

A372245 Triangular array T(n,k) read by rows: column k is the expansion of e.g.f: exp(-2*x)*(exp(x)-1)^k/(2-exp(x)).

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A373940 Expansion of e.g.f. 1/(1 - (exp(x) - 1)^5).

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PARI

PARI

PARI

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A372245 Triangular array T(n,k) read by rows: column k is the expansion of e.g.f: exp(-2x)(exp(x)-1)^k/(2-exp(x)).