A336195
a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^3 * a(k).
Original entry on oeis.org
1, 1, 9, 271, 19353, 2699251, 650553183, 248978967973, 142238892608025, 115699539306013867, 129097362200437841259, 191726066802105786953113, 369666963359241578736653775, 906204961889202975320635813201, 2774573804997927027583123365125685
Offset: 0
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a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^3 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]
nmax = 14; CoefficientList[Series[1/(1 - Sum[x^k/(k!)^3, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^3
A212858
Number of 5 X n arrays with rows being permutations of 0..n-1 and no column j greater than column j-1 in all rows.
Original entry on oeis.org
1, 1, 31, 7291, 7225951, 21855093751, 164481310134301, 2675558106868421881, 84853928323286139485791, 4849446032811641059203617551, 469353176282647626764795665676281, 73159514984813223626195834388445570381, 17619138865526260905773841471696025142373661
Offset: 0
Some solutions for n=3:
2 0 1 0 1 2 0 2 1 0 2 1 1 2 0 0 2 1 2 0 1
2 0 1 2 1 0 0 1 2 0 2 1 0 1 2 1 2 0 2 0 1
0 1 2 2 0 1 0 2 1 2 1 0 0 1 2 0 1 2 2 1 0
2 0 1 0 1 2 1 2 0 0 2 1 1 0 2 2 1 0 1 0 2
1 2 0 0 2 1 2 1 0 1 2 0 0 1 2 2 1 0 2 1 0
Cf.
A000012,
A000225,
A000275,
A212850,
A212851,
A212852,
A212853,
A212854,
A212856,
A212857,
A212859,
A212860,
A336197.
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A212858 := proc(n) sum(z^k/k!^5, k = 0..infinity);
series(%^x, z=0, n+1): n!^5*coeff(%,z,n); add(abs(coeff(%,x,k)), k=0..n) end:
seq(A212858(n), n=1..12); # Peter Luschny, May 27 2017
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T[n_, k_] := T[n, k] = If[k == 0, 1, -Sum[Binomial[k, j]^n*(-1)^j*T[n, k - j], {j, 1, k}]];
a[n_] := T[5, n];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Apr 01 2024, after Alois P. Heinz in A212855 *)
A336196
a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^4 * a(k).
Original entry on oeis.org
1, 1, 17, 1459, 395793, 262131251, 359993423843, 915919888063853, 3975467425523532305, 27639424688447366285203, 292886774320942590679779267, 4544030770812055230064359134573, 99847457331663057820508375752459491, 3021907600842518917755426740899056448141
Offset: 0
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a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^4 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
nmax = 13; CoefficientList[Series[1/(1 - Sum[x^k/(k!)^4, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^4
A336261
a(0) = 1; a(n) = (n!)^5 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^5.
Original entry on oeis.org
1, 1, 33, 8294, 8790208, 28436662624, 228929520628448, 3983602580423420352, 135150778123405293748224, 8262821715336263175482769408, 855516444430388524429593124012032, 142657102263368111456587968163250896896, 36753801552552818015956675623665562408714240
Offset: 0
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b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-i)/i^5, i=1..n))
end:
a:= n-> n!^5*b(n):
seq(a(n), n=0..14); # Alois P. Heinz, Jan 04 2024
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a[0] = 1; a[n_] := a[n] = (n!)^5 Sum[a[k]/(k! (n - k))^5, {k, 0, n - 1}]; Table[a[n], {n, 0, 12}]
nmax = 12; CoefficientList[Series[1/(1 - PolyLog[5, x]), {x, 0, nmax}], x] Range[0, nmax]!^5
A342199
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k)^5 * a(k-1).
Original entry on oeis.org
1, 1, 33, 519, 43111, 5068111, 840782023, 291086377719, 139698959369111, 90748115988081551, 90809507057803456103, 124011515918275951611959, 217278911997171247450862041, 509237348184229328050319432621, 1567286639251140454692258569881053
Offset: 0
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a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^5 a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 14}]
Showing 1-5 of 5 results.
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