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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A360657 Number triangle T associated with 2-Stirling numbers and Lehmer-Comtet numbers (see Comments and Formula section).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 9, 5, 1, 0, 64, 37, 9, 1, 0, 625, 369, 97, 14, 1, 0, 7776, 4651, 1275, 205, 20, 1, 0, 117649, 70993, 19981, 3410, 380, 27, 1, 0, 2097152, 1273609, 365001, 64701, 7770, 644, 35, 1, 0, 43046721, 26269505, 7628545, 1388310, 174951, 15834, 1022, 44, 1
Offset: 0

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Author

Werner Schulte, Feb 15 2023

Keywords

Comments

Triangle T is created using 2-Stirling numbers of the first (A049444) and the second (A143494) kind. The unusual construction is as follows:
Define A(n, k) by recurrence A(n, k) = A(n-1, k-1) + (k+1) * A(n-1, k) for 0 < k < n with initial values A(n, n) = 1, n >= 0, and A(n, 0) = 0, n > 0. A without column k = 0 is A143494. Let B = A^(-1) matrix inverse of A. B without column k = 0 is A049444. Now define T(m, k) = Sum_{i=0..m-k} B(m-k, i) * A(m-1+i, m-1) for 0 < k <= m = n/2 and T(m, 0) = 0^m for 0 <= m = n/2; T(i, j) = 0 if i < j or j < 0.
Matrix inverse of T is A360753. - Werner Schulte, Feb 21 2023
Conjecture: the transpose of this array is the upper triangular matrix U in the LU factorization of the array of Stirling numbers of the second kind read as a square array; the corresponding lower triangular array L is the triangle of Stirling numbers of the second kind. See the example section below. - Peter Bala, Oct 10 2023

Examples

			Triangle T(n, k), 0 <= k <= n, starts:
n\k :  0         1         2        3        4       5      6     7   8  9
==========================================================================
  0 :  1
  1 :  0         1
  2 :  0         2         1
  3 :  0         9         5        1
  4 :  0        64        37        9        1
  5 :  0       625       369       97       14       1
  6 :  0      7776      4651     1275      205      20      1
  7 :  0    117649     70993    19981     3410     380     27     1
  8 :  0   2097152   1273609   365001    64701    7770    644    35   1
  9 :  0  43046721  26269505  7628545  1388310  174951  15834  1022  44  1
  etc.
From _Peter Bala_, Oct 10 2023: (Start)
LU factorization of the square array of Stirling numbers of the second kind (apply Xu, Lemma 2.2):
 / 1               \ / 1   1   1   1  ...\    / 1   1   1    1  ... \
 | 1   1           ||      2   5   9  ...|   |  1   3   6   10  ... |
 | 1   3   1       ||          9  37  ...| = |  1   7  25   65  ... |
 | 1   7   6   1   ||             64  ...|   |  1  15  90  350  ... |
 | ...             ||                 ...|   |  ...                 |
(End)

Links

Crossrefs

Cf. A049444, A143494, A354794, A354795, A088956, A094587.
Cf. A000007 (column 0), A000169 (column 1), A055869 (column 2).
Cf. A000012 (main diagonal), A000096 (1st subdiagonal), A360753 (matrix inverse).

Programs

  • PARI
    tabl(m) = {my(n=2*m, A = matid(n), B, T); for( i = 2, n, for( j = 2, i, A[i, j] = A[i-1, j-1] + j * A[i-1, j] ) ); B = A^(-1); T = matrix( m, m, i, j, if( j == 1, 0^(i-1), sum( r = 0, i-j, B[i-j+1, r+1] * A[i-1+r, i-1] ) ) ); }

Formula

For the definition of triangle T see Comments section.
Conjectured formulas:
1. T(n, k) = (Sum_{i=k..n} A354794(n, i) * (i-1)!) / (k-1)! for 0 < k <= n.
2. T(n, k) - k * T(n, k+1) = A354794(n, k) for 0 <= k <= n.
3. T(n, 1) = A000169(n) = n^(n-1) for n > 0.
4. T(n, 2) = A055869(n-1) = n^(n-1) - (n-1)^(n-1) for n > 1.
5. T(n, k) = (Sum_{i=0..k-1} (-1)^i * binomial(k-1, i) * (n-i)^(n-1)) / (k-1)! for 0 < k <= n.
6. Sum_{i=1..n} (-1)^(n-i) * binomial(n-1+k, i-1) * T(n, i) * (i-1)! = (k-1)^(n-1) for n > 0 and k >= 0.
7. Matrix product of A354795 and T without column 0 equals A094587.
8. Matrix product of T and A354795 without column 0 equals A088956.
9. E.g.f. of column k > 0: Sum_{n>=k} T(n, k) * t^(n-1) / (n-1)! = (W(-t)/(-t)) * (Sum_{n>=k} A354794(n, k) * t^(n-1) / (n-1)!) where W is the Lambert_W-function.

A347940 Array T(n, k) = Sum_{j=2..n+2} (-1)^(n-j)*Stirling2(n+1, j-1)*j!*j^k/2, for n and k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 4, 7, 4, 8, 23, 23, 8, 16, 73, 115, 73, 16, 32, 227, 533, 533, 227, 32, 64, 697, 2359, 3451, 2359, 697, 64, 128, 2123, 10133, 20753, 20753, 10133, 2123, 128, 256, 6433, 42655, 118843, 164731, 118843, 42655, 6433, 256, 512, 19427, 177053, 657833, 1220657, 1220657, 657833, 177053, 19427, 512
Offset: 0

Views

Author

Michel Marcus, Sep 20 2021

Keywords

Comments

T(m, n) is the number of saturated Cp^m*q^n-transfer systems where Cp^m*q^n is the cyclic group of order p^m*q^n, for m, n >= 0, p and q primes. See Hafeez et al. link page 1.

Examples

			Array begins:
   1   2     4      8      16       32 ...
   2   7    23     73     227      697 ...
   4  23   115    533    2359    10133 ...
   8  73   533   3451   20753   118843 ...
  16 227  2359  20753  164731  1220657 ...
  32 697 10133 118843 1220657 11467387 ...
  ...

Links

Crossrefs

Columns k=0-1 gives A000079, A083313(n+1).
Main diagonal gives A220181(n+1).
Cf. A008277 (Stirling2), A143494.

Programs

  • PARI
    T(n, k) = sum(j=2, n+2, (-1)^(n-j)*stirling(n+1, j-1, 2)*j!*j^k/2);

Formula

T(n,k) = T(k,n).
T(n,k) = Sum_{j=0..min(n,k)} (j!*(j+2)!/2)*Stirling2(n+2,j+2;2)*Stirling2(k+2,j+2;2), n,k >= 0, where Stirling2(n,k;2) are the 2-Stirling numbers of the second kind A143494. - Fabián Pereyra, Jan 08 2022
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