A383234 -id:A383234 - cp's OEIS frontend

A383233 Expansion of e.g.f. f(x)^3 * log(f(x)), where f(x) = 1/(1 - 5*x)^(1/5).

0, 1, 11, 167, 3318, 81930, 2423208, 83582568, 3295488816, 146241365904, 7214605476480, 391735046081664, 23216763331632384, 1491431668108800768, 103230214859003968512, 7659080261784464808960, 606407304545822037952512, 51033731719180664212641792, 4549228202963725560906891264

Offset: 0

Seiichi Manyama, Apr 20 2025

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..355

Cf. A383231, A383232, A383234.

With[{nn=20,f=Surd[1/(1-5x),5]},CoefficientList[Series[f^3 Log[f],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 20 2025 *)

a(n) = sum(k=1, n, k*3^(k-1)*5^(n-k)*abs(stirling(n, k, 1)));

a(n) = Sum_{k=1..n} k * 3^(k-1) * 5^(n-k) * |Stirling1(n,k)|.
a(n) = 5^(n-1) * n! * Sum_{k=0..n-1} (-1)^k * binomial(-3/5,k)/(n-k).
a(n) = (10*n-9) * a(n-1) - (5*n-7)^2 * a(n-2) for n > 1.

A383231 Expansion of e.g.f. f(x) * log(f(x)), where f(x) = 1/(1 - 5*x)^(1/5).

Original entry on oeis.org

0, 1, 7, 83, 1394, 30330, 810756, 25710012, 943434288, 39324264624, 1835297984160, 94813760519136, 5371462318747392, 331125138305434368, 22065681276731119104, 1580617232453691210240, 121117633854691036502016, 9885823380533972300470272, 856279708828545483688808448

Offset: 0

Seiichi Manyama, Apr 20 2025

nonn

Cf. A383232, A383233, A383234.

Cf. A004041, A024216, A024382.

a(n) = sum(k=1, n, k*5^(n-k)*abs(stirling(n, k, 1)));

a(n) = Sum_{k=1..n} k * 5^(n-k) * |Stirling1(n,k)|.
a(n) = 5^(n-1) * n! * Sum_{k=0..n-1} (-1)^k * binomial(-1/5,k)/(n-k).
a(n) = (10*n-13) * a(n-1) - (5*n-9)^2 * a(n-2) for n > 1.

A383232 Expansion of e.g.f. f(x)^2 * log(f(x)), where f(x) = 1/(1 - 5*x)^(1/5).

Original entry on oeis.org

0, 1, 9, 122, 2242, 52180, 1471692, 48790608, 1859539344, 80109265824, 3849497255520, 204138860091264, 11842095171021696, 745962168915065088, 50708105952635996928, 3699802551156676392960, 288399758863879774476288, 23919432333548949807869952, 2103184085769044913951461376

Offset: 0

Seiichi Manyama, Apr 20 2025

nonn

Cf. A383231, A383233, A383234.

a(n) = sum(k=1, n, k*2^(k-1)*5^(n-k)*abs(stirling(n, k, 1)));

a(n) = Sum_{k=1..n} k * 2^(k-1) * 5^(n-k) * |Stirling1(n,k)|.
a(n) = 5^(n-1) * n! * Sum_{k=0..n-1} (-1)^k * binomial(-2/5,k)/(n-k).
a(n) = (10*n-11) * a(n-1) - (5*n-8)^2 * a(n-2) for n > 1.

cp's OEIS Frontend

A383233 Expansion of e.g.f. f(x)^3 * log(f(x)), where f(x) = 1/(1 - 5*x)^(1/5).

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A383231 Expansion of e.g.f. f(x) * log(f(x)), where f(x) = 1/(1 - 5*x)^(1/5).

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A383232 Expansion of e.g.f. f(x)^2 * log(f(x)), where f(x) = 1/(1 - 5*x)^(1/5).

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