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A155826 Triangle T(n, k) = binomial(n, k) + binomial(k*(n-k), n) + 2*(-1)^n*StirlingS1(n, k)*StirlingS1(n, n-k), read by rows. 2
4, 1, 1, 1, 4, 1, 1, 15, 15, 1, 1, 76, 249, 76, 1, 1, 485, 3516, 3516, 485, 1, 1, 3606, 46623, 101354, 46623, 3606, 1, 1, 30247, 617541, 2388107, 2388107, 617541, 30247, 1, 1, 282248, 8416315, 51483931, 91651662, 51483931, 8416315, 282248, 1, 1, 2903049, 119667766, 1071669632, 3021085118, 3021085118, 1071669632, 119667766, 2903049, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

FORMULA

T(n, k) = binomial(n, k) + binomial(k*(n-k), n) + 2*(-1)^n*StirlingS1(n, k) * StirlingS1(n, n-k).

Sum_{k=0..n} T(n, k) = 2^n + 2*342111(n) + Sum_{k=0..n} binomial(k*(n-k), n). - G. C. Greubel, Jun 03 2021

EXAMPLE

Triangle begins as:

4;

1, 1;

1, 4, 1;

1, 15, 15, 1;

1, 76, 249, 76, 1;

1, 485, 3516, 3516, 485, 1;

1, 3606, 46623, 101354, 46623, 3606, 1;

1, 30247, 617541, 2388107, 2388107, 617541, 30247, 1;

1, 282248, 8416315, 51483931, 91651662, 51483931, 8416315, 282248, 1;

MATHEMATICA

T[n_, k_]:= Binomial[n, k] + Binomial[k*(n-k), n] + 2*(-1)^n*StirlingS1[n, k]*StirlingS1[n, n-k];

Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 03 2021 *)

PROG

(Magma)

A155826:= func< n, k | Binomial(n, k) + Binomial(k*(n-k), n) + 2*(-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k) >;

[A155826(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 03 2021

(Sage)

def A155826(n, k): return binomial(n, k) + binomial(k*(n-k), n) + 2*stirling_number1(n, k)*stirling_number1(n, n-k)

flatten([[A155826(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 03 2021

CROSSREFS

Cf. A048994, A342111.

Sequence in context: A276330 A234957 A327939 * A273711 A340227 A351942

Adjacent sequences: A155823 A155824 A155825 * A155827 A155828 A155829

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Jan 28 2009

EXTENSIONS

Edited by G. C. Greubel, Jun 03 2021

STATUS

approved

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Last modified January 27 09:55 EST 2023. Contains 359838 sequences. (Running on oeis4.)