A111579 -id:A111579

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page 1

A111673

Triangle, generated from A111579.

+20
2

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 15, 11, 4, 1, 1, 1, 52, 49, 19, 5, 1, 1, 1, 203, 257, 109, 29, 6, 1, 1, 1, 877, 1539, 742, 201, 41, 7, 1, 1, 1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1, 1, 21147, 75905, 51193, 15821, 3176, 505, 71, 9, 1, 1, 1, 115975, 609441, 498118, 170389, 35451, 5497, 729, 89, 10, 1, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

COMMENTS

Columns are inverse binomial transforms of columns (k>0) of A111579.

LINKS

Table of n, a(n) for n=0..77.

A. Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321.

A. Kerber, A matrix of combinatorial numbers related to the symmetric groups, Discrete Math., 21 (1978), 319-321. [Annotated scanned copy]

EXAMPLE

First few rows of the triangle are:

1, 1,

1, 1, 1,

1, 2, 1, 1,

1, 5, 3, 1, 1,

1, 15, 11, 4, 1, 1,

1, 52, 49, 19, 5, 1, 1,

1, 203, 257, 109, 29, 6, 1, 1,

1, 877, 1539, 742, 201, 41, 7, 1, 1,

1, 4140, 10299, 5815, 1657, 331, 55, 8, 1, 1,

...

Inverse binomial transform of column 2 of A111579 (1, 2, 5, 15, 52, 203...) = column 2 (1, 1, 2, 5, 15, 52...).

CROSSREFS

Cf. A111579, A008277.

For two other versions of this triangle see A241578, A241579.

Rows and columns give A000110, A004211, A004212, A004213, A005011, A005012, A028387, A241577.

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Aug 14 2005

EXTENSIONS

More terms from N. J. A. Sloane, Apr 29 2014

STATUS

approved

A111669

Triangle read by rows, based on a simple Fibonacci recursion rule.

+10
4

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 11, 7, 1, 1, 5, 26, 32, 12, 1, 1, 6, 57, 122, 92, 20, 1, 1, 7, 120, 423, 582, 252, 33, 1, 1, 8, 247, 1389, 3333, 2598, 681, 54, 1, 1, 9, 502, 4414, 18054, 24117, 11451, 1815, 88, 1, 1, 10, 1013, 13744, 94684, 210990, 172980, 49566, 4807, 143, 1

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

Subdiagonal is A000071(n+3). Row sums of inverse are 0^n.

Row sums are given by A135934. - Emanuele Munarini, Dec 05 2017

LINKS

Michel Marcus, Rows n = 0..20, flattened

FORMULA

T(n, k) = T(n-1, k-1) + F(k+1)*T(n-1, k) where F(n)=A000045(n).

Column k has g.f. x^k/Product_{j=0..k} (1 - F(j+1)*x).

EXAMPLE

Triangle begins

1....1....2....3....5....8...13....F(k+1)

1....1

1....2....1

1....3....4....1

1....4...11....7....1

1....5...26...32...12....1

1....6...57..122...92...20....1

For example, T(6,3) = 122 = 26 + 3*32 = T(5,2) + F(4)*T(5,3).

MATHEMATICA

(* To generate the triangle *)

Grid[RecurrenceTable[{F[n, k] == F[n-1, k-1] + Fibonacci[k+1] F[n-1, k], F[0, k] == KroneckerDelta[k]}, F, {n, 0, 10}, {k, 0, 10}]] (* Emanuele Munarini, Dec 05 2017 *)

PROG

(PARI) T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, T(n-1, k-1) + fibonacci(k+1)*T(n-1, k))); \\ Michel Marcus, May 25 2024

CROSSREFS

Cf. A111577, A111578, A111579, A008277, A039755, A135934.