A000481 -id:A000481 - cp's OEIS frontend

A346844 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^5 / 5!.

1, 21, 287, 3290, 34671, 350889, 3492511, 34669734, 346231886, 3497726232, 35872743270, 374387203190, 3982122624117, 43207791878715, 478532965417529, 5411213661200830, 62482405993313229, 736696756305382411, 8868148033487285103, 108969560832001750716

Offset: 5

Ilya Gutkovskiy, Aug 05 2021

nonn

Cf. A000110, A000389, A000481, A003128, A005493, A049020, A094816, A346842, A346843.

b:= proc(n, m) option remember;
     `if`(n=0, binomial(m, 5), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=5..24);  # Alois P. Heinz, Aug 05 2021

nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] (Exp[x] - 1)^5/5!, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
Table[Sum[StirlingS2[n, k] Binomial[k, 5], {k, 0, n}], {n, 5, 24}]
Table[Sum[Binomial[n, k] StirlingS2[k, 5] BellB[n - k], {k, 0, n}], {n, 5, 24}]
Table[(-BellB[n] + 89*BellB[n+1] - 145*BellB[n+2] + 75*BellB[n+3] - 15*BellB[n+4] + BellB[n+5])/120, {n, 5, 24}] (* Vaclav Kotesovec, Aug 06 2021 *)

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^5/5!)) \\ Michel Marcus, Aug 06 2021

a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(k,5).
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,5) * Bell(n-k).
a(n) = (-Bell(n) + 89*Bell(n+1) - 145*Bell(n+2) + 75*Bell(n+3) - 15*Bell(n+4) + Bell(n+5))/120. - Vaclav Kotesovec, Aug 06 2021

A056281 Number of primitive (aperiodic) word structures of length n which contain exactly five different symbols.

Original entry on oeis.org

0, 0, 0, 0, 1, 15, 140, 1050, 6951, 42524, 246730, 1379385, 7508501, 40074895, 210766919, 1096189500, 5652751651, 28958088579, 147589284710, 749206047975, 3791262568261, 19137821665325, 96416888184100

Offset: 1

Marks R. Nester

nonn

Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Column 5 of A137651.

Cf. A056270.

a(n) = Sum_{n > 0, d|n} mu(d)*A000481(n/d).

G.f.: Sum_{k>=1} mu(k) * x^(5*k) / Product_{j=1..5} (1 - j*x^k). - Ilya Gutkovskiy, Apr 15 2021

A056474 Number of palindromic structures using exactly five different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 15, 15, 140, 140, 1050, 1050, 6951, 6951, 42525, 42525, 246730, 246730, 1379400, 1379400, 7508501, 7508501, 40075035, 40075035, 210766920, 210766920, 1096190550

Offset: 1

Marks R. Nester

Permuting the symbols will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Index entries for linear recurrences with constant coefficients, signature (1,14,-14,-71,71,154,-154,-120,120).

Cf. A000481, A056470.

Table[StirlingS2[Floor[(n+1)/2],5],{n,40}] (* Harvey P. Dale, Dec 18 2012 *)

stirling2( [(n+1)/2], 5).

G.f.: -x^9/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)). [Colin Barker, Jul 24 2012]

A373173 Triangle read by rows: the exponential almost-Riordan array ( exp(exp(x)-1) | exp(x), exp(x)-1 ).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 1, 3, 1, 15, 1, 7, 6, 1, 52, 1, 15, 25, 10, 1, 203, 1, 31, 90, 65, 15, 1, 877, 1, 63, 301, 350, 140, 21, 1, 4140, 1, 127, 966, 1701, 1050, 266, 28, 1, 21147, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1, 115975, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1

Offset: 0

Stefano Spezia, May 26 2024

			The triangle begins:
    1;
    1, 1;
    2, 1,  1;
    5, 1,  3,  1;
   15, 1,  7,  6,  1;
   52, 1, 15, 25, 10,  1;
  203, 1, 31, 90, 65, 15, 1;
  ...

Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 14.

Cf. A000012 (k=1), A000225, A000392 (k=3), A000453 (k=4), A000481 (k=5), A000770 (k=6), A000771 (k=7), A049394 (k=8), A049435 (k=10), A049447 (k=9).

Triangle A008277 with 1st column A000110.

T[n_,0]:=n!SeriesCoefficient[Exp[Exp[x]-1],{x,0,n}]; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[Exp[x](Exp[x]-1)^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

T(n,0) = n! * [x^n] exp(exp(x)-1); T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] exp(x)*(exp(x)-1)^(k-1).

T(n,2) = A000225(n-1) for n > 1.

cp's OEIS Frontend

A346844 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^5 / 5!.

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A056281 Number of primitive (aperiodic) word structures of length n which contain exactly five different symbols.

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A056474 Number of palindromic structures using exactly five different symbols.

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A373173 Triangle read by rows: the exponential almost-Riordan array ( exp(exp(x)-1) | exp(x), exp(x)-1 ).

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