A346889
Expansion of e.g.f. 1 / (1 - x^3 * exp(x) / 3!).
Original entry on oeis.org
1, 0, 0, 1, 4, 10, 40, 315, 2296, 15204, 117720, 1127445, 11531740, 120909646, 1370809804, 17111895255, 227853866800, 3182209445640, 47003318806896, 737325061500009, 12187616610231540, 210930852047426770, 3821604062633503300, 72479758506840597451
Offset: 0
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nmax = 23; CoefficientList[Series[1/(1 - x^3 Exp[x]/3!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 3] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
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my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^3*exp(x)/3!))) \\ Michel Marcus, Aug 06 2021
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a(n) = n!*sum(k=0, n\3, k^(n-3*k)/(6^k*(n-3*k)!)); \\ Seiichi Manyama, May 13 2022
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
A351703
Square array T(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - x^k * exp(x) / k!).
Original entry on oeis.org
1, 1, 1, 1, 0, 4, 1, 0, 1, 21, 1, 0, 0, 3, 148, 1, 0, 0, 1, 12, 1305, 1, 0, 0, 0, 4, 70, 13806, 1, 0, 0, 0, 1, 10, 465, 170401, 1, 0, 0, 0, 0, 5, 40, 3591, 2403640, 1, 0, 0, 0, 0, 1, 15, 315, 31948, 38143377, 1, 0, 0, 0, 0, 0, 6, 35, 2296, 319068, 672552730
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
1, 0, 0, 0, 0, 0, ...
4, 1, 0, 0, 0, 0, ...
21, 3, 1, 0, 0, 0, ...
148, 12, 4, 1, 0, 0, ...
1305, 70, 10, 5, 1, 0, ...
13806, 465, 40, 15, 6, 1, ...
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T(n, k) = if(n==0, 1, binomial(n, k)*sum(j=0, n-k, binomial(n-k, j)*T(j, k)));
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T(n, k) = n!*sum(j=0, n\k, j^(n-k*j)/(k!^j*(n-k*j)!)); \\ Seiichi Manyama, May 13 2022
A346893
Expansion of e.g.f. 1 / (1 - x^5 * exp(x) / 5!).
Original entry on oeis.org
1, 0, 0, 0, 0, 1, 6, 21, 56, 126, 504, 6006, 67320, 577863, 4038034, 24975951, 165481680, 1553590220, 19495772856, 249507077436, 2910465717648, 31103684847837, 326286335505438, 3766644374319673, 51399738264984648, 785038533451101930
Offset: 0
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nmax = 25; CoefficientList[Series[1/(1 - x^5 Exp[x]/5!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 5] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]
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my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^5*exp(x)/5!))) \\ Michel Marcus, Aug 06 2021
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a(n) = n!*sum(k=0, n\5, k^(n-5*k)/(120^k*(n-5*k)!)); \\ Seiichi Manyama, May 13 2022
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 1-5 of 5 results.