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A176155 Regular triangle, T(n, k) = f(n, k) - f(n, 0) + 1, where f(n, k) = Sum_{j=0..k} StirlingS1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} StirlingS1(n, n-j)*binomial(n, j), read by rows. 5
1, 1, 1, 1, -1, 1, 1, -8, -8, 1, 1, -23, 67, -23, 1, 1, -49, 181, 181, -49, 1, 1, -89, 1906, -6704, 1906, -89, 1, 1, -146, -1511, 9808, 9808, -1511, -146, 1, 1, -223, 49113, -426551, 782671, -426551, 49113, -223, 1, 1, -323, -343547, 3220453, -3873389, -3873389, 3220453, -343547, -323, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Row sum are: {1, 2, 1, -14, 23, 266, -3068, 16304, 27351, -1993610, 31213301, ...}.

LINKS

G. C. Greubel, Rows n = 0..100 of triangle, flattened

FORMULA

With f(n, k) = Sum_{j=0..k} StirlingS1(n, n-j)*binomial(n,j) + Sum_{j=0..n-k} StirlingS1(n, n-j)*binomial(n, j) then T(n, k) = f(n, k) - f(n, 0) + 1.

EXAMPLE

Triangle begins as:

  1;

  1,    1;

  1,   -1,       1;

  1,   -8,      -8,       1;

  1,  -23,      67,     -23,        1;

  1,  -49,     181,     181,      -49,        1;

  1,  -89,    1906,   -6704,     1906,      -89,       1;

  1, -146,   -1511,    9808,     9808,    -1511,    -146,       1;

  1, -223,   49113, -426551,   782671,  -426551,   49113,    -223,    1;

  1, -323, -343547, 3220453, -3873389, -3873389, 3220453, -343547, -323, 1;

MAPLE

with(combinat);

f:= proc(n, k) option remember; add(stirling1(n, n-j)*binomial(n, j), j=0..k) + add(stirling1(n, n-j)* binomial(n, j), j=0..n-k); end;

seq(seq(f(n, k) -f(n, 0) +1, k=0..n), n=0..10); # G. C. Greubel, Nov 26 2019

MATHEMATICA

f[n_, k_]:= Sum[StirlingS1[n, n-j]*Binomial[n, j], {j, 0, k}] + Sum[StirlingS1[n, n-j]*Binomial[n, j], {j, 0, n-k}];

Table[f[n, k] - f[n, 0] + 1, {n, 0, 10}, {k, 0, n}]//Flatten

PROG

(PARI)

f(n, k) = sum(j=0, k, stirling(n, n-j, 1)*binomial(n, j)) + sum(j=0, n-k, stirling(n, n-j, 1)*binomial(n, j));

T(n, k) = f(n, k) - f(n, 0) + 1; \\ G. C. Greubel, Nov 26 2019

(MAGMA)

f:= func< n, k | (&+[StirlingFirst(n, n-j)*Binomial(n, j): j in [0..k]]) + (&+[StirlingFirst(n, n-j)*Binomial(n, j): j in [0..n-k]]) >;

[f(n, k) - f(n, 0) + 1: k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 26 2019

(Sage)

def f(n, k): return sum((-1)^j*stirling_number1(n, n-j)*binomial(n, j) for j in (0..k)) + sum((-1)^j*stirling_number1(n, n-j)*binomial(n, j) for j in (0..n-k))

[[f(n, k)-f(n, 0)+1 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 26 2019

(GAP)

f:= function(n, k) return Sum([0..k], j-> (-1)^j*Stirling1(n, n-j)*Binomial(n, j)) + Sum([0..n-k], j-> (-1)^j*Stirling1(n, n-j)*Binomial(n, j)); end;

Flat(List([0..10], n-> List([0..n], k-> f(n, k)-f(n, 0)+1 ))); # G. C. Greubel, Nov 26 2019

CROSSREFS

Cf. A048994, A132393, A176153, A176154, A176156, A176157.

Sequence in context: A110940 A172352 A141134 * A174127 A230153 A091648

Adjacent sequences:  A176152 A176153 A176154 * A176156 A176157 A176158

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula, Apr 10 2010

EXTENSIONS

Name edited by G. C. Greubel, Nov 27 2019

STATUS

approved

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Last modified July 31 18:30 EDT 2021. Contains 346376 sequences. (Running on oeis4.)