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
A346890
Expansion of e.g.f. 1 / (1 - x^4 * exp(x) / 4!).
Original entry on oeis.org
1, 0, 0, 0, 1, 5, 15, 35, 140, 1386, 12810, 92730, 589545, 4234945, 41832791, 483334215, 5401798220, 57262207380, 626438655900, 7740130412796, 107197808258745, 1546730804858085, 22360919412385015, 329241486278715395, 5121840342205301946
Offset: 0
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nmax = 24; CoefficientList[Series[1/(1 - x^4 Exp[x]/4!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k, 4] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]
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my(x='x+O('x^25)); Vec(serlaplace(1/(1-x^4*exp(x)/4!))) \\ Michel Marcus, Aug 06 2021
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a(n) = n!*sum(k=0, n\4, k^(n-4*k)/(24^k*(n-4*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
A370986
Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^2/2*exp(x)) ).
Original entry on oeis.org
1, 0, 1, 3, 24, 220, 2535, 35931, 588028, 11110212, 236355885, 5605425595, 146586048906, 4190560545678, 130037568893323, 4352949788266275, 156362710268877960, 5999465853656510056, 244887352806333261753, 10595948826118665719475, 484448170190529051468910
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^2/2*exp(x)))/x))
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a(n) = sum(k=0, n\2, k^(n-2*k)*(n+k)!/(2^k*k!*(n-2*k)!))/(n+1);
Showing 1-5 of 5 results.