A225470 -id:A225470 - cp's OEIS frontend

A383222 Coefficient of x^4 in expansion of (x+2) * (x+5) * ... * (x+3*n-1).

0, 0, 0, 0, 1, 40, 1275, 39655, 1276009, 43382934, 1570298610, 60630265740, 2495678898636, 109326548645600, 5085420626585936, 250576924194171120, 13046999027750243984, 716156618057417103008, 41347880768363832470304, 2505655766070932929630464

Offset: 0

Seiichi Manyama, Apr 20 2025

Column k=4 of A225470.

a(n) = polcoef(prod(k=0, n-1, x+3*k+2), 4);

first(n) = {my(res = vector(n), v = [1, 0, 0, 0, 0], cv, c = -1, pc = 1); for(i = 2, n, c+=3; pc *= c; cv = v[^5]; cv = concat(0, cv); cv+=v*c; v = cv; res[i] = v[5]); res} \\ David A. Corneth, May 06 2025

a(n) = Sum_{k=4..n} 2^(k-4) * 3^(n-k) * binomial(k,4) * |Stirling1(n,k)|.
E.g.f.: f(x)^2 * log(f(x))^4 / 24, where f(x) = 1/(1 - 3*x)^(1/3).
a(n) = Sum_{k=4..n} (3*n-1)^(k-4) * 3^(n-k) * binomial(k,4) * Stirling1(n,k). - Seiichi Manyama, May 06 2025

A225474 Triangle read by rows, k!2^ks_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 2, 3, 8, 8, 15, 46, 72, 48, 105, 352, 688, 768, 384, 945, 3378, 7600, 11040, 9600, 3840, 10395, 39048, 97112, 167040, 193920, 138240, 46080, 135135, 528414, 1418648, 2754192, 3857280, 3736320, 2257920, 645120, 2027025, 8196480, 23393376, 49824768, 79892736

Offset: 0

Peter Luschny, May 19 2013

The Stirling-Frobenius cycle numbers are defined in A225470.

			[n\k][ 0,    1,    2,     3,    4,    5]
[0]    1,
[1]    1,    2,
[2]    3,    8,    8,
[3]   15,   46,   72,    48,
[4]  105,  352,  688,   768,  384,
[5]  945, 3378, 7600, 11040, 9600, 3840.

Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

Cf. A028338, A225479 (m=1), A048594, A161198, A225475.

Cf. A000079, A000142, A000165, A001147, A014479, A028338.

SFCSO[n_, k_, m_] := SFCSO[n, k, m] = If[k>n || k<0, 0, If[n == 0 && k == 0, 1, m*k*SFCSO[n-1, k-1, m] + (m*n-1)*SFCSO[n-1, k, m]]]; Table[SFCSO[n, k, 2], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 05 2014, translated from Sage *)

@CachedFunction
def SF_CSO(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return m*k*SF_CSO(n-1, k-1, m) + (m*n-1)*SF_CSO(n-1, k, m)
for n in (0..8): [SF_CSO(n, k, 2) for k in (0..n)]

For a recurrence see the Sage program.
T(n, 0) ~ A001147; T(n, n) ~ A000165; T(n, n-1) ~ A014479.
T(n,k) = A028338(n,k) * A000165(k) = A225475(n,k) * A000079(k) = A161198(n,k) * A000142(k). - Philippe Deléham, Jun 25 2015

cp's OEIS Frontend

A383222 Coefficient of x^4 in expansion of (x+2) * (x+5) * ... * (x+3*n-1).

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A225474 Triangle read by rows, k!2^ks_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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cp's OEIS Frontend

A383222 Coefficient of x^4 in expansion of (x+2) * (x+5) * ... * (x+3*n-1).

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A225474 Triangle read by rows, k!*2^k*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A225474 Triangle read by rows, k!2^ks_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.