The On-Line Encyclopedia of Integer Sequences (OEIS)

Revisions by Tani Akinari (See also Tani Akinari's wiki page)

(Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A071722

Expansion of (1+x^2*C^2)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
(history; published version)

#8 by Tani Akinari at Sun Dec 01 18:31:17 EST 2024

STATUS

editing

proposed

#7 by Tani Akinari at Sun Dec 01 18:30:30 EST 2024

FORMULA

a(n) = (3*binomial(2*n+2,n)+5*binomial(2*n,n+2))/(n+3). - Tani Akinari, Dec 01 2024

PROG

(Maxima) a(n):=(3*binomial(2*n+2, n)+5*binomial(2*n, n+2))/(n+3); makelist(a(n), n, 0, 50);

/* Tani Akinari, Dec 01 2024 */

STATUS

approved

editing

A071723

Expansion of (1+x^2*C^2)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
(history; published version)

#13 by Tani Akinari at Sun Dec 01 17:56:02 EST 2024

STATUS

editing

proposed

#12 by Tani Akinari at Sun Dec 01 17:55:25 EST 2024

FORMULA

a(n) = (4*binomial(2*n+3,n)+6*binomial(2*n+1,n+3))/(n+4). - Tani Akinari, Dec 01 2024

PROG

(Maxima) a(n):=(4*binomial(2*n+3, n)+6*binomial(2*n+1, n+3))/(n+4); /* Tani Akinari, Dec 01 2024 */

STATUS

approved

editing

A071727

Expansion of (1+x^3*C)*C^2, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
(history; published version)

#9 by Tani Akinari at Thu Nov 28 07:10:00 EST 2024

STATUS

editing

proposed

Discussion

Thu Nov 28

07:16

Alois P. Heinz: not in Maple ... there binomial(-2,-2) gives 1, not 0 ...

07:23

Tani Akinari: in Maxima, binomial(-2,-2) gives 0, not 1.

07:26

Alois P. Heinz: yes, I guessed ...

#5 by Tani Akinari at Thu Nov 28 05:07:32 EST 2024

STATUS

editing

proposed

Discussion

Thu Nov 28

06:39

Alois P. Heinz: gives 5 for n=1 ...

#4 by Tani Akinari at Thu Nov 28 05:04:09 EST 2024

FORMULA

For n > 0, a(n) = 3*binomial(2*n-4,n-3)/n+2*binomial(2*n+1,n)/(n+2). - Tani Akinari, Nov 28 2024

PROG

(Maxima) a(n):=if n=0 then 1 else 3*binomial(2*n-4, n-3)/n+2*binomial(2*n+1, n)/(n+2);

makelist(a(n), n, 0, 50); /* Tani Akinari, Nov 28 2024 */

STATUS

approved

editing

A077230

Numerators of coefficients of series expansion of a certain integral in the theory of charged particle beams.
(history; published version)

#16 by Tani Akinari at Tue Nov 05 15:49:37 EST 2024

STATUS

editing

proposed

#15 by Tani Akinari at Tue Nov 05 15:47:11 EST 2024

FORMULA

a(n) = numerator(Sum_{k=0..n} binomial(1/2-n, k)*binomial(2*n-1/2, n-k)*Stirling2(n+k, k)*k!/((1-2*n)*(n+k)!)), n>0, a(0)=2. - Tani Akinari, Nov 05 2024

PROG

(Maxima) a(n):=if n=0 then 2 else num(sum(binomial(1/2-n, k)*binomial(2*n-1/2, n-k)*stirling2(n+k, k)*k!/((1-2*n)*(n+k)!), k, 0, n)); /* Tani Akinari, Nov 05 2024 */

STATUS

approved

editing

A242233

2^n*(C_n)^(1/2) in the Cauchy type product where C_n is the n-th Catalan number.
(history; published version)

#13 by Tani Akinari at Tue Nov 05 14:31:11 EST 2024

STATUS

editing

proposed