A081048 -id:A081048 - cp's OEIS frontend

A344640 Antidiagonal sums of A344639.

1, 2, 5, 16, 63, 297, 1649, 10641, 78823, 662315, 6241889, 65294039, 751035233, 9420926879, 127958645921, 1870319380463, 29263787708393, 487891616911031, 8632986776222945, 161555987833199807, 3187606376603319017, 66128414623822131863, 1438861202348688524897, 32763278185929878499887

Offset: 0

Stefano Spezia, May 25 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..448

Cf. A000254, A081048, A344639.

Table[Sum[Sum[Abs[StirlingS1[n-k,m]](m+1)^k,{m,0,n-k}],{k,0,n}],{n,0,23}]

a(n) = Sum_{k=0..n} Sum_{m=0..n-k} abs(S1(n-k, m))*(m + 1)^k, where S1 indicates the signed Stirling numbers of first kind.

Conjecture: a(n) ~ n!. - Vaclav Kotesovec, May 26 2021

A345651 Fourth column of A008296.

Original entry on oeis.org

1, 10, 25, -35, 49, 0, -820, 9020, -87164, 859144, -8965320, 100136400, -1199838576, 15406135488, -211479420096, 3094582896000, -48129022468224, 793274283938304, -13818265424460288, 253731538514893824, -4899371564756837376, 99261476593521868800

Offset: 4

Luca Onnis, Aug 26 2021

sign

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, b(n,4).

Cf. A008296, A045406, A081048.

Cf. A001008, A002805, A007406, A007407.

b:= proc(n, k) option remember; `if`(n=k, 1, `if`(k=0, 0,
      (n-1)*b(n-2, k-1)+b(n-1, k-1)+(k-n+1)*b(n-1, k)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..28);  # Alois P. Heinz, Aug 26 2021
# alternative
seq(A008296(n,4),n=4..70) ; # R. J. Mathar, Sep 15 2021

a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n - 2)!;
a[n_, n_] = 1;
a[n_, k_] := a[n, k] = (n - 1) a[n - 2, k - 1] +
    a[n - 1, k - 1] + (k - n + 1) a[n - 1, k];
Flatten[Table[N[a[n + 4, 4], 10], {n, 1, 400}]]

a(n) = sum(m=4, n, binomial(m, 4)*4^(m-4)*stirling(n, m, 1)); \\ Michel Marcus, Sep 14 2021

a(n) = A008296(n,4).
a(n) = (-1)^n*(4*H(n-5,1)^3 + 8*H(n-5,3) - 12*H(n-5,2)*H(n-5,1) - 25*H(n-5,1)^2 + 25*H(n-5,2) + 35*H(n-5,1) - 10)*(n-5)! for n >= 5 where H(n,1) = Sum_{j=1..n} 1/j is the n-th harmonic number, H(n,2) = Sum_{j=1..n} 1/j^2 and H(n,3) = Sum_{j=1..n} 1/j^3.
a(n) = Sum_{m=4..n} binomial(m,4) * 4^(m-4) * Stirling1(n,m). - Alois P. Heinz, Aug 26 2021
Conjecture: D-finite with recurrence a(n) +2*(2*n-13)*a(n-1) +(6*n^2-84*n+295)*a(n-2) +(2*n-15)*(2*n^2-30*n+113)*a(n-3) +(n-8)^4*a(n-4)=0. - R. J. Mathar, Sep 15 2021

A347276 Third column of A008296.

Original entry on oeis.org

1, 6, 5, -15, 49, -196, 944, -5340, 34716, -254760, 2078856, -18620784, 180973584, -1887504768, 20887922304, -242111586816, 2889841121280, -34586897978880, 393722260047360, -3659128846433280, 5687630494110720, 1137542166526464000, -49644151627682304000

Offset: 3

Luca Onnis, Aug 25 2021

sign

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, b(n,3).

Cf. A008296, A045406, A081048.
Cf. A001008, A002805, A007406, A007407.
Cf. A048994, A008275.

b:= proc(n, k) option remember; `if`(n=k, 1, `if`(k=0, 0,
      (n-1)*b(n-2, k-1)+b(n-1, k-1)+(k-n+1)*b(n-1, k)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..30);  # Alois P. Heinz, Aug 25 2021

a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n - 2)!;
a[n_, n_] = 1;
a[n_, k_] :=  a[n, k] = (n - 1) a[n - 2, k - 1] +
    a[n - 1, k - 1] + (k - n + 1) a[n - 1, k];
Flatten[Table[a[n + 3, 3], {n, 0, 30}]]

a(n) = sum(m=3, n, binomial(m, 3)*3^(m-3)*stirling(n, m, 1)); \\ Michel Marcus, Sep 14 2021

a(n) = A008296(n,3).
a(n) = (-1)^n*(3*H(n-4,1)^2 - 3*H(n-4,2) - 11*H(n-4,1) + 6)*(n-4)! for n >= 4, where H(n,1) = Sum_{j=1..n} 1/j = A001008(n)/A002805(n) is the n-th harmonic number and H(n,2) = Sum_{j=1..n} 1/j^2 = A007406(n)/A007407(n).
a(n) = Sum_{m=3..n} binomial(m,3) * 3^(m-3) * Stirling1(n,m). - Alois P. Heinz, Aug 26 2021

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A344640 Antidiagonal sums of A344639.

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A345651 Fourth column of A008296.

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A347276 Third column of A008296.

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