A015001 -id:A015001 - cp's OEIS frontend

A347486 Irregular triangle read by rows: T(n, k) is the q-multinomial coefficient defined by the k-th partition of n in Abramowitz-Stegun order, evaluated at q = 3.

1, 1, 4, 1, 13, 52, 1, 40, 130, 520, 2080, 1, 121, 1210, 4840, 15730, 62920, 251680, 1, 364, 11011, 33880, 44044, 440440, 1431430, 1761760, 5725720, 22902880, 91611520, 1, 1093, 99463, 925771, 397852, 12035023, 37030840, 120350230, 48140092, 481400920, 1564552990

Offset: 1

Álvar Ibeas, Sep 03 2021

Abuse of notation: we write T(n, L) for T(n, k), where L is the k-th partition of n in A-St order.

For any permutation (e_1,...,e_r) of the parts of L, T(n, L) is the number of chains of subspaces 0 < V_1 < ··· < V_r = (F_3)^n with dimension increments (e_1,...,e_r).

			The number of subspace chains 0 < V_1 < V_2 < (F_3)^3 is 52 = T(3, (1, 1, 1)). There are 13 = A022167(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 4 = A022167(2, 1) extensions to a two-dimensional subspace V_2.
Triangle begins:
  k:  1   2    3    4     5     6      7
      ----------------------------------
n=1:  1
n=2:  1   4
n=3:  1  13   52
n=4:  1  40  130  520  2080
n=5:  1 121 1210 4840 15730 62920 251680

R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.

Álvar Ibeas, First 20 rows, flattened

Cf. A036038 (q = 1), A022167, A015001 (last entry in each row).

T(n, (n)) = 1. T(n, L) = A022167(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.

A015015 q-factorial numbers for q=-3.

Original entry on oeis.org

1, 1, -2, -14, 280, 17080, -3108560, -1700382320, 2788627004800, 13722833490620800, -202576467988544249600, -8971504037808659182035200, 1191954026463258458925196672000, 475090227821752019816863814722432000, -568085339196037403679856371543830284544000, -2037851067068183667490280132124059680133919488000

Offset: 0

Olivier Gérard

Vincenzo Librandi, Table of n, a(n) for n = 0..60
Index entries for sequences related to factorial numbers.

Column k=3 of A384454.

Cf. A015001.

[n le 1 select 1 else ((-3)^n - 1)*Self(n-1)/(-4): n in [1..18]]; // Vincenzo Librandi, Oct 26 2012

RecurrenceTable[{a[1]==1, a[n]==(((-3)^n - 1) * a[n-1])/(-4)}, a, {n, 18}] (* Vincenzo Librandi, Oct 26 2012 *)

a(n) = Product_{k=1..n} ((-3)^k - 1)/(-3 - 1).
a(1) = 1, a(n) = ((-3)^n - 1)*a(n-1)/(-4). - Vincenzo Librandi, Oct 26 2012
a(n) ~ (-1)^floor(n/2) * c * 3^(n*(n+1)/2) / 4^n, where c = Product_{k>=1} (1 - 1/(-3)^k) = 1.2176479365615020492... . - Amiram Eldar, Aug 09 2025

a(0)=1 prepended by Seiichi Manyama, May 30 2025

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A347486 Irregular triangle read by rows: T(n, k) is the q-multinomial coefficient defined by the k-th partition of n in Abramowitz-Stegun order, evaluated at q = 3.

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A015015 q-factorial numbers for q=-3.

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