A264753 Irregular triangle read by rows: T(n,k) = A127671(n,k)/A036040(n,k), n >= 1 and 1 <= k <= A000041(n).

1, 1, -1, 1, -1, 2, 1, -1, -1, 2, -6, 1, -1, -1, 2, 2, -6, 24, 1, -1, -1, -1, 2, 2, 2, -6, -6, 24, -120, 1, -1, -1, -1, 2, 2, 2, 2, -6, -6, -6, 24, 24, -120, 720, 1, -1, -1, -1, -1, 2, 2, 2, 2, 2, -6, -6, -6, -6, -6, 24, 24, 24, -120, -120, 720, -5040

Offset

1,6

Comments

This sequence connects the multinomial coefficients A036040 (M_3) with A127671 (M_5).

The numbers of terms in n-th row is the number of partitions A000041(n). The number of terms T(n, k) with equal values in the n-th row follow the rhythm of A008284(n).

Some row sums are [1, 0, 2, -5, 21, -104, 636, -4511, 36455, -330954, 3334390, -36914039].

Links

Table of n, a(n) for n=1..66.

Milton Abramowitz and Irene A. Stegun, editors, Multinomials: M_1, M_2 and M_3, Handbook of Mathematical Functions, December 1972, pp. 831-2.

Formula

T(n, k) = A127671(n, k)/A036040(n, k), n >= 1 and 1 <= k <= A000041(n).

Example

The first few rows of the T(n,k) triangle:
n = 1: 1
n = 2: 1, -1
n = 3: 1, -1, 2
n = 4: 1, -1, -1, 2, -6
n = 5: 1, -1, -1, 2, 2, -6, 24
n = 6: 1, -1, -1, -1, 2, 2, 2, -6, -6, 24, -120
n = 7: 1, -1, -1, -1, 2, 2, 2, 2, -6, -6, -6, 24, 24, -120, 720

Maple

nmax:=8: with(combinat): A008284 := proc(n, k) option remember; if k < 0 or n < 0 then 0 elif k = 0 then if n = 0 then 1 else 0 fi else A008284(n-1, k-1) + A008284(n-k, k) fi end: for n from 1 to nmax do p:=0: k:=1: while k < numbpart(n)+1 do p := p+1: k1 := A008284(n, p): while k1 > 0 do A264753(n, k) := (-1)^(p+1)*(p-1)!: k := k+1: k1 := k1-1: od: od: od: seq(seq(A264753(n, k), k = 1..numbpart(n)), n = 1..nmax);

Mathematica

nMax = 8; A008284[n_, k_] := A008284[n, k] = If[k<0 || n<0, 0, If[k == 0, If[n == 0, 1, 0], A008284[n-1, k-1] + A008284[n-k, k]]]; For[n = 1, n <= nMax, n++, p = 0; k = 1; While[k < PartitionsP[n]+1, p = p+1; k1 = A008284[n, p]; While[k1>0, A264753[n, k] = (-1)^(p+1)*(p-1)!; k = k+1; k1 = k1-1]]]; Table[Table[A264753[n, k], {k, 1, PartitionsP[n]}], {n, 1, nMax}] // Flatten (* Jean-François Alcover, Oct 01 2016, translated from Maple *)

Crossrefs

Cf. A036040 (M_3), A127671 (M_5), A000041, A008284, A081362.

Cf. A048996 (M_0), A036038 (M_1), A036039 (M_2), A117506 (M_4).

Sequence in context: A221131 A126886 A179272 * A165680 A248049 A231867

Adjacent sequences: A264750 A264751 A264752 * A264754 A264755 A264756

Keyword

sign,tabf

Author

Johannes W. Meijer, Jul 12 2016

Status

approved