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A123217 Triangle read by rows: the n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A123162(n,j)*x^j*(1 - x)^(n - j). 12
1, 1, 1, 1, -1, 1, 2, 3, -5, 1, 3, 20, -32, 9, 1, 4, 58, -82, 5, 15, 1, 5, 125, -108, -161, 170, -31, 1, 6, 229, 17, -797, 603, 7, -65, 1, 7, 378, 532, -2210, 664, 1468, -968, 129, 1, 8, 580, 1820, -4226, -2846, 8788, -4388, 9, 255, 1, 9, 843, 4440, -5262 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

LINKS

Table of n, a(n) for n=0..58.

FORMULA

From Franck Maminirina Ramaharo, Oct 10 2018: (Start)

Row n = coefficients in the expansion of (1 - x)^n + x*((1 - 2*sqrt((1 - x)*x))^n*(1 - x + sqrt((1 - x)*x)) - (1 - x - sqrt((1 - x)*x))*(1 + 2*sqrt((1 - x)*x))^n)/ (2*sqrt((1 - x)*x)*(2*x - 1)).

G.f.: (1 - (2 - x)*y + (1 - 4*x + 3*x^2)*y^2 - (x - 3*x^2 + 2*x^3)*y^3)/(1 - (3 - x)*y + (3 - 6*x + 4*x^2)*y^2 - (1 - 5*x + 8*x^2 - 4*x^3)*y^3).

E.g.f.: exp((1 - x)*y) + x*((1 - x + sqrt((1 - x)*x))*exp((1 - 2*sqrt((1 - x)*x))*y) - (1 - x - sqrt((1 - x)*x))*exp((1 + 2*sqrt((1 - x)*x))*y))/(2*(2*x - 1)*sqrt((1 - x)*x)) - (1 - 3*x)/(1 - 2*x) + 1. (End)

EXAMPLE

Triangle begins:

     1;

     1;

     1, 1,  -1;

     1, 2,   3,   -5;

     1, 3,  20,  -32,     9;

     1, 4,  58,  -82,     5,    15;

     1, 6, 229,   17,  -797,   603,    7,   -65;

     1, 7, 378,  532, -2210,   664, 1468,  -968, 129;

     1, 8, 580, 1820, -4226, -2846, 8788, -4388,   9, 255;

     ... reformatted and extended. - Franck Maminirina Ramaharo, Oct 10 2018

MATHEMATICA

t[n_, m_] = If [n == m == 0 || m == 0, 1, (2*n - 1)!/((2*(n - m))!*(2*m - 1)!)];

a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];

v = Table[CoefficientList[Sum[a[[n + 1]][[m + 1]]*x^ m*(1 - x)^(n - m), {m, 0, n}], x], {n, 0, 10}]; Flatten[v]

PROG

(Maxima) A123162(n, k) := if n = 0 and k = 0 or k = 0 then 1 else binomial(2*n - 1, 2*k - 1)$

P(x, n) := expand(sum(A123162(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$

T(n, k) := ratcoef(P(x, n), x, k)$

tabf(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, hipow(P(x, n), x)))$ /* Franck Maminirina Ramaharo, Oct 10 2018 */

CROSSREFS

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123202, A123221.

Sequence in context: A295302 A110879 A016587 * A039704 A002752 A239693

Adjacent sequences:  A123214 A123215 A123216 * A123218 A123219 A123220

KEYWORD

tabf,sign

AUTHOR

Roger L. Bagula, Oct 04 2006

EXTENSIONS

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 11 2018

STATUS

approved

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Last modified June 23 11:29 EDT 2021. Contains 345397 sequences. (Running on oeis4.)