A212857
Number of 4 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, 15, 1135, 271375, 158408751, 191740223841, 429966316953825, 1644839120884915215, 10079117505143103766735, 94135092186827772028779265, 1287215725538576868883610346465, 24929029117106417518788960414909025, 664978827664071363541997348802227351425
Offset: 0
Some solutions for n=3:
1 2 0 1 0 2 1 0 2 2 1 0 2 0 1 2 1 0 1 0 2
2 1 0 1 0 2 0 2 1 0 2 1 2 1 0 1 0 2 2 1 0
1 2 0 2 1 0 1 0 2 0 1 2 2 1 0 2 1 0 1 2 0
2 1 0 0 1 2 2 1 0 2 1 0 1 0 2 2 1 0 2 1 0
Cf.
A000012,
A000225,
A000275,
A212850,
A212851,
A212852,
A212853,
A212854,
A212856,
A212858,
A212859,
A212860,
A336196.
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A212857 := proc(n) sum(z^k/k!^4, k = 0..infinity);
series(%^x, z=0, n+1): n!^4*coeff(%,z,n); add(abs(coeff(%,x,k)), k=0..n) end:
seq(A212857(n), n=1..13); # 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[4, n];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Apr 01 2024, after Alois P. Heinz in A212855 *)
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
A336197
a(0) = 1; a(n) = Sum_{k=0..n-1} binomial(n,k)^5 * a(k).
Original entry on oeis.org
1, 1, 33, 8263, 8718945, 28076306251, 224968772934303, 3896175006605313013, 131557135159637950535265, 8004845815916146011992853811, 824857614282973828473497207276283, 136888961901974254918775560412316183913, 35099479542762449254288789631427310686677535
Offset: 0
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a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^5 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 12}]
nmax = 12; CoefficientList[Series[1/(1 - Sum[x^k/(k!)^5, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!^5
A336260
a(0) = 1; a(n) = (n!)^4 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^4.
Original entry on oeis.org
1, 1, 17, 1474, 404768, 271581776, 377987513392, 974814164752800, 4289222350867156992, 30232332223815625555968, 324796212685273837095714816, 5108947647642107040382284423168, 113818571142935411070742114448769024, 3492592855002964381945529723625305210880
Offset: 0
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b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-i)/i^4, i=1..n))
end:
a:= n-> n!^4*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!)^4 Sum[a[k]/(k! (n - k))^4, {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
nmax = 13; CoefficientList[Series[1/(1 - PolyLog[4, x]), {x, 0, nmax}], x] Range[0, nmax]!^4
A342198
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k)^4 * a(k-1).
Original entry on oeis.org
1, 1, 17, 179, 6083, 298583, 20015947, 2214261035, 332014246747, 64923646898023, 17220997162396851, 5898373172881341811, 2513698997312409032785, 1335813901379210302030497, 875400777321767437156156305, 692119702624591542667897216641, 653524900495231808524498551469617
Offset: 0
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a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]^4 a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 16}]
Showing 1-5 of 5 results.
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