A211228 -id:A211228 - cp's OEIS frontend

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

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.

A211227 Row sums of A211226.

Original entry on oeis.org

1, 2, 3, 4, 8, 8, 20, 16, 48, 32, 112, 64, 256, 128, 576, 256, 1280, 512, 2816, 1024, 6144, 2048, 13312, 4096, 28672, 8192, 61440, 16384, 131072, 32768, 278528, 65536, 589824, 131072, 1245184, 262144, 2621440, 524288, 5505024

Offset: 0

Peter Bala, Apr 05 2012

The odd-indexed terms of the sequence a(2*n-1) count the compositions of n+1, while the even-indexed terms a(2*n) count the total number of parts in the composition of n+1. Compare with A211228.

			The four compositions of 3 are 1+1+1, 1+2, 2+1 and 3 having 8 parts in total. Hence a(3) = 4 and a(4) = 8.

Cf. A000079, A001792, A211226.

a(n) = sum {k = 0..n } f(n)/(f(k)*f(n-k)), where f(n) := (floor(n/2))!.
a(2*n-1) = 2^n = A000079(n); a(2*n) = (n+2)*2^(n-1) = A001792(n).
O.g.f.: (1+2*x-x^2-4*x^3)/(1-2*x^2)^2 = 1 + 2*x + 3*x^2 + 4*x^3 + 8*x^4 + ....
E.g.f.: cosh(sqrt(2)*x) + (4+x)/(2*sqrt(2))*sinh(sqrt(2)*x) = 1 + 2*x + 3*x^2/2! + 4*x^3/3! + 8*x^4/4! + .....

cp's OEIS Frontend

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

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