A143398
Triangle T(n,k) = number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains k labels, n>=0, 0<=k<=n.
Original entry on oeis.org
1, 0, 1, 0, 3, 1, 0, 10, 3, 1, 0, 41, 9, 4, 1, 0, 196, 40, 10, 5, 1, 0, 1057, 210, 30, 15, 6, 1, 0, 6322, 1176, 175, 35, 21, 7, 1, 0, 41393, 7273, 1176, 105, 56, 28, 8, 1, 0, 293608, 49932, 7084, 756, 126, 84, 36, 9, 1, 0, 2237921, 372060, 42120, 6510, 378, 210, 120, 45, 10, 1
Offset: 0
T(4,2) = 9: 3->{1,2}<-4, 2->{1,3}<-4, 2->{1,4}<-3, 1->{2,3}<-4, 1->{2,4}<-3, 1->{3,4}<-2, {1,2}{3,4}, {1,3}{2,4}, {1,4}{2,3}.
Triangle begins:
1;
0, 1;
0, 3, 1;
0, 10, 3, 1;
0, 41, 9, 4, 1;
0, 196, 40, 10, 5, 1;
...
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u:= (n, k)-> `if`(k=0, 0, floor(n/k)):
T:= (n, k)-> n! *add(i^(n-k*i)/ ((n-k*i)! *i! *k!^i), i=0..u(n, k)):
seq(seq(T(n, k), k=0..n), n=0..12);
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t[n_, n_] = 1; t[, 0] = 0; t[n, k_] := n!*Sum[i^(n-k*i)/((n-k*i)!*i!*k!^i), {i, 0, n/k}]; Table[t[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 19 2013 *)
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u(n,k) = if(k==0, 0, n\k);
T(n, k) = n!*sum(j=0, u(n, k), j^(n-k*j)/(k!^j*j!*(n-k*j)!)); \\ Seiichi Manyama, May 13 2022
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
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
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
A355666
Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - x^k/k! * (exp(x) - 1)).
Original entry on oeis.org
1, 1, 1, 1, 0, 3, 1, 0, 2, 13, 1, 0, 0, 3, 75, 1, 0, 0, 3, 28, 541, 1, 0, 0, 0, 6, 125, 4683, 1, 0, 0, 0, 4, 10, 1146, 47293, 1, 0, 0, 0, 0, 10, 195, 8827, 545835, 1, 0, 0, 0, 0, 5, 20, 1281, 94200, 7087261, 1, 0, 0, 0, 0, 0, 15, 35, 5908, 1007001, 102247563, 1, 0, 0, 0, 0, 0, 6, 35, 1176, 68076, 12814390, 1622632573
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 0, 0, 0, 0, 0, 0, ...
3, 2, 0, 0, 0, 0, 0, ...
13, 3, 3, 0, 0, 0, 0, ...
75, 28, 6, 4, 0, 0, 0, ...
541, 125, 10, 10, 5, 0, 0, ...
4683, 1146, 195, 20, 15, 6, 0, ...
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T(n, k) = n!*sum(j=0, n\(k+1), j!*stirling(n-k*j, j, 2)/(k!^j*(n-k*j)!));
Showing 1-6 of 6 results.