A154715 -id:A154715

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A258773

Triangle read by rows, T(n,k) = (-1)^(n-k)*C(n,k)*k^n, for n >= 0 and 0 <= k <= n.

+10
4

1, 0, 1, 0, -2, 4, 0, 3, -24, 27, 0, -4, 96, -324, 256, 0, 5, -320, 2430, -5120, 3125, 0, -6, 960, -14580, 61440, -93750, 46656, 0, 7, -2688, 76545, -573440, 1640625, -1959552, 823543, 0, -8, 7168, -367416, 4587520, -21875000, 47029248, -46118408, 16777216

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

OFFSET

0,5

COMMENTS

The row polynomials are p(0, x) = 1, and p(n, x) = Eu(x)^n (x-1)^n, for n >= 1, where Eu(x) := x*d/dx is the Euler derivative with respect to x. See A075513. - Wolfdieter Lang, Oct 12 2022

Coefficients of the Sidi polynomials (-1)^n*x*D_{n,0,n}(x), for n >= 0, where D_{k,n,m}(z) is given in Theorem 4.2., p. 862, of Sidi [1980]. - Wolfdieter Lang, Apr 10 2023

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1326

Avram Sidi, Numerical Quadrature and Nonlinear Sequence Transformations; Unified Rules for Efficient Computation of Integrals with Algebraic and Logarithmic Endpoint Singularities, Math. Comp., 35 (1980), 851-874.

FORMULA

Sum_{k=0..n} T(n,k) = n!.

Sum_{k=0..n} |T(n,k)| = A072034(n).

Sum_{n>=0} Sum_{k=0..n} T(n,k) x^k y^n/n! = 1/(1 + W(-x*y*exp(-y)) where W is the Lambert W function. - Robert Israel, Dec 16 2015

T(n,n) = A000312(n). - Peter Luschny, Dec 17 2015

T(n, k+1) = n * A075513(n, k) if n > 0. - Michael Somos, May 13 2018

EXAMPLE

[1]

[0, 1]

[0, -2, 4]

[0, 3, -24, 27]

[0, -4, 96, -324, 256]

[0, 5, -320, 2430, -5120, 3125]

[0, -6, 960, -14580, 61440, -93750, 46656]

[0, 7, -2688, 76545, -573440, 1640625, -1959552, 823543]

MAPLE

seq(seq((-1)^(n-k)*binomial(n, k)*k^n, k=0..n), n=0..8);

T_row := proc(n) (-1)^n*(1-exp(x))^n/n!; diff(%, [x$n]); subs(exp(x)=t, n!*expand(%, x)); CoefficientList(%, t) end: seq(print(T_row(n)), n=0..7);

MATHEMATICA

Flatten@Table[(-1)^(n - k) Binomial[n, k] k^n, {n, 0 , 10}, {k, 0, n}] (* G. C. Greubel, Dec 17 2015 *)

CROSSREFS

Cf. A000142, A000312, A072034, A075513, A154715.

KEYWORD

sign,easy,tabl

AUTHOR

Peter Luschny, Jun 09 2015

STATUS

approved

A362353

Triangle read by rows: T(n,k) = (-1)^(n-k)*binomial(n, k)*(k+3)^n, for n >= 0, and k = 0,1, ..., n. Coefficients of certain Sidi polynomials.

+10
1

1, -3, 4, 9, -32, 25, -27, 192, -375, 216, 81, -1024, 3750, -5184, 2401, -243, 5120, -31250, 77760, -84035, 32768, 729, -24576, 234375, -933120, 1764735, -1572864, 531441, -2187, 114688, -1640625, 9797760, -28824005, 44040192, -33480783, 10000000, 6561, -524288, 10937500, -94058496, 403536070, -939524096, 1205308188, -800000000, 214358881

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

OFFSET

0,2

COMMENTS

This is the member N = 2 of a family of signed triangles with row sums n! = A000142(n): T(N; n, k) = (-1)^(n-k)*binomial(n, k)*(k + N + 1)^n, for integer N, n >= 0 and k = 0, 1, ..., n. The row polynomials PS(N; n, z) = Sum_{k=0..n} T(N; n, k)*z^k = ((-1)^n/z^N)*D_{n,N+1,n}(z) in [Sidi 1980].

For N = -1, 0 and 1 see A258773(n, k), A075513(n+1, k) and (-1)^(n-k) * A154715(n, k), respectively.

The column sequences, for k = 0, 1, ..., 6 and n >= k, are A141413(n+2), (-1)^(n+1)*A018215(n) = 4*(-1)^(n+1)*A002697(n), 5^2*(-1)^n*A081135(n), (-1)^(n+1)*A128964(n-1) = 6^3*(-1)^(n+1)*A081144(n), 7^4*(-1)^n*A139641(n-4), 2^15*(-1)^(n+1)*A173155(n-5), 3^12*(-1)^n*A173191(n-6), respectively.

The e.g.f. of the triangle (see below) needs the exponential convolution (LambertW(-z)/(-z))^2 = Sum_{n>=0} c(2; n)*z^n/n!, where c(2; n) = Sum_{m=0..n} |A137352(n+1, m)|*2^m = A007334(n+2).

The row sums give n! = A000142(n).

LINKS

Paolo Xausa, Table of n, a(n) for n = 0..5049 (rows 0..100 of the triangle, flattened)

Wolfdieter Lang, On a Certain Family of Sidi Polynomials, May 2023.

FORMULA

T(n, k) = (-1)^(n-k)*binomial(n, k)*(k + 3)^n, for n >= 0, k = 0, 1, ..., n.

O.g.f. of column k: (x*(k + 3))^k/(1 - (k + 3)*x)^(k+1), for k >= 0.

E.g.f. of column k: exp(-(k + 3)*x)*((k + 3)*x)^k/k!, for k >= 0.

E.g.f. of the triangle, that is, the e.g.f. of its row polynomials {PS(2;n,y)}_{n>=0}): ES(2;y,x) = exp(-3*x)*(1/3)*(d/dz)(W(-z)/(-z))^2, after replacing z by x*y*exp(-x), where W is the Lambert W-function for the principal branch. This becomes ES(2;y,x) = exp(-3*x)*exp(3*(-W(-z)))/(1 - (-W(-z)), with z = x*y*exp(-x).

EXAMPLE

The triangle T begins:

n\k 0 1 2 3 4 5 6 7

0: 1

1: -3 4

2: 9 -32 25

3: -27 192 -375 216

4: 81 -1024 3750 -5184 2401

5: -243 5120 -31250 77760 -84035 32768

6: 729 -24576 234375 -933120 1764735 -1572864 531441

7: -2187 114688 -1640625 9797760 -28824005 44040192 -33480783 10000000

...

n = 8: 6561 -524288 10937500 -94058496 403536070 -939524096 1205308188 -800000000 2143588,

n = 9: -19683 2359296 -70312500 846526464 -5084554482 16911433728 -32543321076 36000000000 -21221529219 5159780352.

MATHEMATICA

A362353row[n_]:=Table[(-1)^(n-k)Binomial[n, k](k+3)^n, {k, 0, n}]; Array[A362353row, 10, 0] (* Paolo Xausa, Jul 30 2023 *)

CROSSREFS

Cf. A000142 (row sums), A075513, A154715, A258773.

Columns k = 0..6 involve (see above): A002697, A007334, A018215, A081135, A081144, A128964, A137352, A139641, A141413, A173155, A173191.

KEYWORD

sign,tabl,easy

AUTHOR

Harlan J. Brothers and Wolfdieter Lang, Apr 27 2023

EXTENSIONS

a(41)-a(44) from Paolo Xausa, Jul 31 2023

STATUS

approved

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