A211230 -id:A211230 - cp's OEIS frontend

A211231 Row sums of A211230.

1, 3, 6, 10, 27, 33, 108, 108, 405, 351, 1458, 1134, 5103, 3645, 17496, 11664, 59049, 37179, 196830, 118098, 649539, 373977, 2125764, 1180980, 6908733, 3720087, 22320522, 11691702, 71744535

Offset: 0

Peter Bala, Apr 05 2012

Cf. A211227, A211230.

a(2*n) = (n+1)*3^n; a(2*n+1) = (n+9)*3^(n-1).

O.g.f.: (1+3*x-8*x^3)/(1-3*x^2)^2 = 1 + 3*x + 6*x^2 + 10*x^3 + ....

A211226 Triangular array: T(n,k) = f(n)/(f(k)*f(n-k)), where f(n) = (floor(n/2))!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 6, 3, 3, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 4, 4, 12, 6, 12, 4, 4, 1, 1, 1, 4, 4, 6, 6, 4, 4, 1, 1, 1, 5, 5, 20, 10, 30, 10, 20, 5, 5, 1, 1, 1, 5, 5, 10, 10, 10, 10, 5, 5, 1, 1, 1, 6, 6, 30, 15

Offset: 0

Peter Bala, Apr 05 2012

			Triangle begins
.n\k.|....0....1....2....3....4....5....6....7....8....9...10...11
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
..0..|....1
..1..|....1....1
..2..|....1....1....1
..3..|....1....1....1....1
..4..|....1....2....2....2....1
..5..|....1....1....2....2....1....1
..6..|....1....3....3....6....3....3....1
..7..|....1....1....3....3....3....3....1....1
..8..|....1....4....4...12....6...12....4....4....1
..9..|....1....1....4....4....6....6....4....4....1....1
.10..|....1....5....5...20...10...30...10...20....5....5....1
.11..|....1....1....5....5...10...10...10...10....5....5....1....1
...

Peter Bala, Notes on A211226

Cf. A007318, A056040, A211227 (row sums), A211228 (shallow diagonal sums), A211229 (inverse), A211230 (array squared).

T(n,k) := f(n)/(f(k)*f(n-k)), where f(n) := (floor(n/2))!.
T(2*n+1,2*k) = T(2*n+1,2*k+1) = T(2*n,2*k) = binomial(n,k);
T(2*n,2*k+1) = n*binomial(n-1,k).
Recurrence equations:
T(2*n,2*k) = T(2*n-1,2*k) + T(2*n-1,2*k-1);
T(2*n,2*k+1) = T(2*n-1,2*k+1) + (n-1)*T(2*n-1,2*k);
T(2*n+1,2*k) = T(2*n,2*k); T(2*n+1,2*k+1) = T(2*n,2*k).
The Star of David property holds:
T(n,k+1)*T(n+1,k)*T(n+2,k+2) = T(n,k)*T(n+2,k+1)*T(n+1,k+2).
O.g.f.: (1 + t*(1+x) - t^2*(1-x+x^2) - t^3*(1+x+x^2+x^3))/(1 - t^2*(1+x^2))^2 = sum {n>=0} R(n,x)*t^n = 1 + (1+x)*t + (1+x+x^2)*t^2 + (1+x+x^2+x^3)*t^3 + ....
E.g.f.: cosh(t*sqrt(1+x^2)) + (1+x+x*t/2)/sqrt(1+x^2)*sinh(t*sqrt(1+x^2)) = sum {n>=0} R(n,x)*t^n/n! = 1 + (1+x)*t + (1+x+x^2)*t^2/2! + (1+x+x^2+x^3)*t^3/3! + ....
Row generating polynomials: R(2*n+1,x) = (1+x)*(1+x^2)^n; R(2*n,x) = (1+n*x+x^2)*(1+x^2)^(n-1).
Row sums: A211227. Shallow diagonal sums A211228. Central terms T(2*n,n) equal A056040(n).
The inverse array A211229 involves the derangement numbers A000166. The squared array is A211230.

cp's OEIS Frontend

A211231 Row sums of A211230.

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