cp's OEIS Frontend

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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A069138 Triangle formed by multiplying Stirling numbers of 2nd kind S2(n,m) (A008277) by m+1.

Original entry on oeis.org

2, 2, 3, 2, 9, 4, 2, 21, 24, 5, 2, 45, 100, 50, 6, 2, 93, 360, 325, 90, 7, 2, 189, 1204, 1750, 840, 147, 8, 2, 381, 3864, 8505, 6300, 1862, 224, 9, 2, 765, 12100, 38850, 41706, 18522, 3696, 324, 10, 2, 1533, 37320, 170525, 255150, 159789, 47040, 6750, 450, 11
Offset: 1

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Author

N. J. A. Sloane, Apr 10 2002

Keywords

Comments

The number of rhyme schemes for a stanza of n+1 lines with m rhyming syllables in its first n lines.

Examples

			Triangle begins:
  2;
  2,  3;
  2,  9,   4;
  2, 21,  24,  5;
  2, 45, 100, 50, 6;
  ...

References

Links

Crossrefs

Row sums give Bell numbers A000110.
Cf. A360174 (Stirling1 counterpart), A360205 (Lah counterpart).

Programs

  • PARI
    T(n, m) = stirling(n, m, 2)*(m+1);
tabl(nn) = for(n=1, nn, for (k=1, n, print1(T(n, m), ", ")); print); \\ Michel Marcus, Sep 21 2017

Formula

T(n, m) = (m+1)*S2(n, m).

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jul 01 2002

A360205 Triangle read by rows. T(n, k) = (-1)^(n-k)*(k+1)*binomial(n, k)*pochhammer(1-n, n-k).

Original entry on oeis.org

1, 0, 2, 0, 4, 3, 0, 12, 18, 4, 0, 48, 108, 48, 5, 0, 240, 720, 480, 100, 6, 0, 1440, 5400, 4800, 1500, 180, 7, 0, 10080, 45360, 50400, 21000, 3780, 294, 8, 0, 80640, 423360, 564480, 294000, 70560, 8232, 448, 9, 0, 725760, 4354560, 6773760, 4233600, 1270080, 197568, 16128, 648, 10
Offset: 0

Views

Author

Peter Luschny, Feb 08 2023

Keywords

Comments

A refinement of the number of partial permutations of an n-set (A002720).
Also the coefficients of a shifted derivative of the unsigned Lah polynomials (A271703).

Examples

			Triangle T(n, k) starts:
[0] 1;
[1] 0,     2;
[2] 0,     4,      3;
[3] 0,    12,     18,      4;
[4] 0,    48,    108,     48,      5;
[5] 0,   240,    720,    480,    100,     6;
[6] 0,  1440,   5400,   4800,   1500,   180,    7;
[7] 0, 10080,  45360,  50400,  21000,  3780,  294,   8;
[8] 0, 80640, 423360, 564480, 294000, 70560, 8232, 448, 9;

Crossrefs

Cf. A052849 (column 1), A045991 (subdiagonal), A002720 (row sums), A271703.
Cf. A069138 (Stirling2 counterpart), A360174 (Stirling1 counterpart).

Programs

  • Maple
    T := (n, k) -> (-1)^(n - k)*(k + 1)*binomial(n, k)*pochhammer(1 - n, n - k):
seq(seq(T(n, k), k = 0..n), n = 0..9);
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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