A375954 -id:A375954 - cp's OEIS frontend

A375948 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^(3/2).

1, 3, 18, 153, 1683, 22698, 362403, 6683463, 139787568, 3269240883, 84535585263, 2394699999948, 73749495626253, 2453332830142743, 87667856626175298, 3349116499958627733, 136209377351085310863, 5875794769594996985778, 267968680043585007829383

Offset: 0

Seiichi Manyama, Sep 03 2024

nonn

Cf. A005649, A375949.
Cf. A004123, A367470, A367471, A375954.
Cf. A001147.

nmax=18; CoefficientList[Series[1 / (3 - 2 * Exp[x])^(3/2),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)

a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k+1)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} A001147(k+1) * Stirling2(n,k).

a(n) ~ 2^(3/2) * n^(n+1) / (3^(3/2) * log(3/2)^(n + 3/2) * exp(n)). - Vaclav Kotesovec, May 20 2025

A375991 Expansion of e.g.f. (3 - 2 * exp(x))^(3/2).

Original entry on oeis.org

1, -3, 0, 9, 45, 252, 1935, 19989, 260190, 4063887, 73823445, 1527002694, 35408499885, 909389617497, 25618701424680, 785355764569749, 26024092206299505, 926859918577582332, 35306305954587340515, 1432301360556686816529, 61649353087003554947550

Offset: 0

Seiichi Manyama, Sep 05 2024

Cf. A004123, A305404, A367470, A375948, A375954.

Cf. A006681.

With[{nn=20},CoefficientList[Series[(3-2Exp[x])^(3/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 19 2025 *)

a(n) = sum(k=0, n, prod(j=0, k-1, 2*j-3)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (2*j-3)) * Stirling2(n,k).

a(n) ~ 3^(5/2) * n^(n-2) / (2^(3/2) * exp(n) * log(3/2)^(n - 3/2)). - Vaclav Kotesovec, May 20 2025

cp's OEIS Frontend

A375948 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^(3/2).

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