A346888
Expansion of e.g.f. 1 / (1 - x^2 * exp(x) / 2).
Original entry on oeis.org
1, 0, 1, 3, 12, 70, 465, 3591, 31948, 319068, 3539385, 43205635, 575312826, 8298867798, 128921967265, 2145837600375, 38097353658120, 718657756980376, 14354000800751313, 302625047150614179, 6716038666999745710, 156498725047355717250, 3820426102008414736761
Offset: 0
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nmax = 22; CoefficientList[Series[1/(1 - x^2 Exp[x]/2), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
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my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^2*exp(x)/2))) \\ Michel Marcus, Aug 06 2021
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a(n) = n!*sum(k=0, n\2, k^(n-2*k)/(2^k*(n-2*k)!)); \\ Seiichi Manyama, May 13 2022
A336960
E.g.f.: 1 / (1 - x * (2 + x) * exp(x)).
Original entry on oeis.org
1, 2, 14, 132, 1676, 26590, 506202, 11242952, 285383240, 8149464954, 258575410190, 9024809281972, 343619185754748, 14173557899208422, 629600469603730562, 29965010056866657600, 1521221783964264806672, 82053967063309770102130, 4686301361507067542636694
Offset: 0
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nmax = 18; CoefficientList[Series[1/(1 - x (2 + x) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k (k + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
A352357
Expansion of e.g.f. 1/(1 - Sum_{k>=1} binomial(k+2,3) * x^k/k!).
Original entry on oeis.org
1, 1, 6, 40, 364, 4155, 56836, 907158, 16547896, 339587445, 7743161740, 194212763756, 5314051343932, 157520046898695, 5028409083962824, 171984217743856890, 6274444932921616176, 243215342466576246185, 9982290554423689511124, 432464578359391409082952
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-(x+x^2+x^3/6)*exp(x))))
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a(n) = if(n==0, 1, sum(k=1, n, binomial(k+2, 3)*binomial(n, k)*a(n-k)));
A352358
Expansion of e.g.f. 1/(1 - Sum_{k>=1} binomial(k+3,4) * x^k/k!).
Original entry on oeis.org
1, 1, 7, 51, 509, 6390, 96036, 1684284, 33760588, 761287221, 19074162865, 525696741801, 15805694091243, 514818296979974, 18058391314446224, 678683621386945560, 27207234575709663516, 1158858397815372736601, 52263672918705232821477
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-(x+3*x^2/2+x^3/2+x^4/24)*exp(x))))
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a(n) = if(n==0, 1, sum(k=1, n, binomial(k+3, 4)*binomial(n, k)*a(n-k)));
Showing 1-4 of 4 results.