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A123162 Triangle read by rows: T(n,k) = binomial(2*n - 1, 2*k - 1) for 0 < k <= n and T(n,0) = 1. 3
1, 1, 1, 1, 3, 1, 1, 5, 10, 1, 1, 7, 35, 21, 1, 1, 9, 84, 126, 36, 1, 1, 11, 165, 462, 330, 55, 1, 1, 13, 286, 1287, 1716, 715, 78, 1, 1, 15, 455, 3003, 6435, 5005, 1365, 105, 1, 1, 17, 680, 6188, 19448, 24310, 12376, 2380, 136, 1, 1, 19, 969, 11628, 50388 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Muniru A Asiru, Table of n, a(n) for n = 0..1325(rows 0 t0 50, flattened)

FORMULA

From Paul Barry, May 26 2008: (Start)

T(n,k) = C(2n - 1, 2k - 1) + 0^k.

Column k has g.f. (x^k/(1 - x)^(2*k + 0^k))*sum{j=0..k} C(2*k, 2*j)*x^j. (End)

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

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

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

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

EXAMPLE

Triangle begins:

     1;

     1,  1;

     1,  3,   1;

     1,  5,  10,    1;

     1,  7,  35,   21,    1;

     1,  9,  84,  126,   36,    1;

     1, 11, 165,  462,  330,   55,    1;

     1, 13, 286, 1287, 1716,  715,   78,  1;

     1, 15, 455, 3003, 6435, 5005, 1365, 105, 1;

     ...

MATHEMATICA

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

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

PROG

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

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

(GAP) Flat(Concatenation([1], List([1..10], n->Concatenation([1], List([1..n], m->Binomial(2*n-1, 2*m-1)))))); # Muniru A Asiru, Oct 11 2018

CROSSREFS

Cf. A007318, A034867, A103327, A122366.

Sequence in context: A201588 A086385 A295222 * A213998 A294946 A083075

Adjacent sequences:  A123159 A123160 A123161 * A123163 A123164 A123165

KEYWORD

nonn,tabl,easy

AUTHOR

Roger L. Bagula, Oct 02 2006

EXTENSIONS

Edited by N. J. A. Sloane, Oct 04 2006

Partially edited and offset corrected by Franck Maminirina Ramaharo, Oct 10 2018

STATUS

approved

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Last modified July 13 04:53 EDT 2020. Contains 335673 sequences. (Running on oeis4.)