A143568 -id:A143568 - cp's OEIS frontend

A143565 Triangle T(n,k), n>=1, 1<=k<=n, where the e.g.f. for column k satisfies: A_k(x) = exp(x*A_k(x^k/k!)).

1, 3, 1, 16, 4, 1, 125, 13, 5, 1, 1296, 46, 21, 6, 1, 16807, 241, 61, 31, 7, 1, 262144, 1471, 211, 106, 43, 8, 1, 4782969, 9409, 1401, 281, 169, 57, 9, 1, 100000000, 67348, 8065, 946, 505, 253, 73, 10, 1, 2357947691, 564841, 37241, 7561, 1261, 841, 361, 91, 11, 1

Offset: 1

Alois P. Heinz, Aug 24 2008

			Triangle begins:
       1;
       3,    1;
      16,    4,    1;
     125,   13,    5,    1;
    1296,   46,   21,    6,    1;
   16807,  241,   61,   31,    7,    1;
  262144, 1471,  211,  106,   43,    8,    1;
  ...

Alois P. Heinz, Rows n = 1..100, flattened

Columns 1-9: A000272, A143566, A143567, A143568, A143569, A143570, A143571, A143572, A143573.

T(2n,n) gives A319924.

A:= proc(n,k::posint) option remember; if n<=0 then 1 else unapply(
      convert(series(exp(x*A(n-k,k)(x^k/k!)), x,n+1), polynom),x) fi
    end:
T:= (n,k)-> coeff(A(n,k)(x), x,n)*n!:
seq(seq(T(n,k), k=1..n), n=1..12);

a[n_, k_] := a[n, k] = If[n <= 0, 1&, Function[x, Series[E^(x*a[n - k, k][x^k/k!]), {x, 0, n+1}] // Normal // Evaluate]]; t[n_, k_] := Coefficient[a[n, k][x], x, n]*n!; Flatten[Table[Table[t[n, k], {k, 1, n}], {n, 1, 12}]] (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

E.g.f. for column k satisfies: A_k(x) = exp(x*A_k(x^k/k!)).

T(0,k) = 1; T(n,k) = (n-1)! * Sum_{j=0..floor((n-1)/k)} (k*j+1) * T(j,k) * T(n-1-k*j,k) / (k!^j * j! * (n-1-k*j)!). - Seiichi Manyama, Nov 28 2023

A354552 Expansion of e.g.f. exp( x * exp(x^4/24) ).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 31, 106, 281, 946, 7561, 54286, 281161, 1207636, 7997991, 81996916, 701522641, 4580581916, 29742355441, 306369616636, 3632198902321, 34710574441096, 276645112305871, 2652825718776696, 35647605796451881, 458142859493786776

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

This sequence is different from A143568.

Seiichi Manyama, Table of n, a(n) for n = 0..533

Cf. A000248, A354550, A354551.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(x^4/24)))))

a(n) = n!*sum(k=0, n\4, (n-4*k)^k/(24^k*k!*(n-4*k)!));

a(n) = n! * Sum_{k=0..floor(n/4)} (n - 4*k)^k/(24^k * k! * (n - 4*k)!).

A367756 E.g.f. satisfies A(x) = exp(x * (1 + x + x^2 + x^3) * A(x^4/24)).

Original entry on oeis.org

1, 1, 3, 13, 73, 386, 2671, 20728, 175393, 1553968, 15520861, 165541806, 1869485773, 22249874518, 284029764383, 3804116563276, 53328350650081, 782331158754088, 12051288543702313, 193028133988081918, 3212490296905001781, 55543932173668760221

Offset: 0

Seiichi Manyama, Nov 29 2023

nonn

Cf. A367754, A367755, A367757.

Cf. A143568.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, i-1, (j+1)*v[j\4+1]*v[i-j]/(24^(j\4)*(j\4)!*(i-1-j)!))); v;

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..n-1} (k+1) * a(floor(k/4)) * a(n-1-k) / (24^floor(k/4) * floor(k/4)! * (n-1-k)!).

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A143565 Triangle T(n,k), n>=1, 1<=k<=n, where the e.g.f. for column k satisfies: A_k(x) = exp(x*A_k(x^k/k!)).

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A354552 Expansion of e.g.f. exp( x * exp(x^4/24) ).

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A367756 E.g.f. satisfies A(x) = exp(x * (1 + x + x^2 + x^3) * A(x^4/24)).

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