A110180 -id:A110180 - cp's OEIS frontend

A110181 Row sums of number triangle A110180.

1, 2, 3, 6, 15, 42, 131, 432, 1497, 5420, 20373, 79422, 319927, 1328764, 5677653, 24904584, 111961129, 515029020, 2421047613, 11616330342, 56829235095, 283211069352, 1436529522233, 7410596379456, 38852645340297

Offset: 0

Paul Barry, Jul 14 2005

a(n)=sum{k=0..n, sum{j=0..floor((n-k)/2), C(n-k, j)C(n-k-j, j)k^j}}.

A110182 Diagonal sums of number triangle A110180.

Original entry on oeis.org

1, 1, 2, 2, 5, 9, 26, 66, 199, 575, 1790, 5542, 17945, 58733, 198880, 685296, 2429281, 8778321, 32491940, 122503412, 471379771, 1844669495, 7346727298, 29712688698, 122054603983, 508548201863, 2149211636944, 9204566602696, 39944636660655

Offset: 0

Paul Barry, Jul 14 2005

a(n)=sum{k=0..floor(n/2), sum{j=0..floor((n-2k)/2), C(n-2k, j)C(n-2k-j, j)k^j}}

A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1-4k)*x^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 7, 1, 1, 1, 7, 13, 19, 1, 1, 1, 9, 19, 49, 51, 1, 1, 1, 11, 25, 91, 161, 141, 1, 1, 1, 13, 31, 145, 331, 581, 393, 1, 1, 1, 15, 37, 211, 561, 1441, 2045, 1107, 1, 1, 1, 17, 43, 289, 851, 2841, 5797, 7393, 3139, 1

Offset: 0

Seiichi Manyama, May 01 2019

			Square array begins:
   1,   1,    1,    1,     1,     1,     1, ...
   1,   1,    1,    1,     1,     1,     1, ...
   1,   3,    5,    7,     9,    11,    13, ...
   1,   7,   13,   19,    25,    31,    37, ...
   1,  19,   49,   91,   145,   211,   289, ...
   1,  51,  161,  331,   561,   851,  1201, ...
   1, 141,  581, 1441,  2841,  4901,  7741, ...
   1, 393, 2045, 5797, 12489, 22961, 38053, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
T. D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Columns k=0..6 give A000012, A002426, A084601, A084603, A084605, A098264, A098265.
Main diagonal gives A187018.
Cf. A110180, A292627, A307847, A307860, A307910.

T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[n, j] * Binomial[n-j, j], {j, 0, Floor[n/2]}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 13 2021 *)

A(n,k) is the coefficient of x^n in the expansion of (1 + x + k*x^2)^n.
A(n,k) = Sum_{j=0..floor(n/2)} k^j * binomial(n,j) * binomial(n-j,j) = Sum_{j=0..floor(n/2)} k^j * binomial(n,2*j) * binomial(2*j,j).
D-finite with recurrence: n * A(n,k) = (2*n-1) * A(n-1,k) - (1-4*k) * (n-1) * A(n-2,k).

cp's OEIS Frontend

A110181 Row sums of number triangle A110180.

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A110182 Diagonal sums of number triangle A110180.

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A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1-4k)*x^2).

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cp's OEIS Frontend

A110181 Row sums of number triangle A110180.

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Author

Keywords

Formula

A110182 Diagonal sums of number triangle A110180.

Views

Author

Keywords

Formula

A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1-4*k)*x^2).

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Author

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Programs

Mathematica

Formula

A307855 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2x + (1-4k)*x^2).