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A211226 Triangular array: T(n,k) = f(n)/(f(k)*f(n-k)), where f(n) = (floor(n/2))!. 8
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,12

LINKS

Table of n, a(n) for n=0..82.

Peter Bala, Notes on A211226

FORMULA

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.

EXAMPLE

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

...

CROSSREFS

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

Sequence in context: A048858 A246465 A172497 * A135265 A144110 A076490

Adjacent sequences:  A211223 A211224 A211225 * A211227 A211228 A211229

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Bala, Apr 05 2012

STATUS

approved

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Last modified October 20 01:39 EDT 2018. Contains 316378 sequences. (Running on oeis4.)