cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Peter Bala, Apr 05 2012

Keywords

Examples

			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
...

Links

Crossrefs

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

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.

A211230 Matrix square of lower triangular array A211226.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 8, 8, 6, 4, 1, 8, 8, 8, 6, 2, 1, 20, 24, 24, 24, 9, 6, 1, 16, 20, 24, 24, 12, 9, 2, 1, 48, 64, 80, 96, 48, 48, 12, 8, 1, 32, 48, 64, 80, 48, 48, 16, 12, 2, 1, 112, 160, 240, 320, 200, 240, 80, 80, 15, 10, 1, 64, 112, 160, 240, 160
Offset: 0

Views

Author

Peter Bala, Apr 05 2012

Keywords

Comments

Analog of square of Pascal's triangle.

Examples

			Triangle begins
.n\k.|....0....1....2....3....4....5....6....7....8....9
= = = = = = = = = = = = = = = = = = = = = = = = = = = = =
..0..|....1
..1..|....2....1
..2..|....3....2....1
..3..|....4....3....2....1
..4..|....8....8....6....4....1
..5..|....8....8....8....6....2....1
..6..|...20...24...24...24....9....6....1
..7..|...16...20...24...24...12....9....2....1
..8..|...48...64...80...96...48...48...12....8....1
..9..|...32...48...64...80...48...48...16...12....2....1
...

Crossrefs

Cf. A211226, A211231 (row sums).

Formula

T(2*n,2*k) = T(2*n+1,2*k+1) = (n+2-k)*binomial(n,k)*2^(n-k-1);
T(2*n,2*k+1) = n*binomial(n-1,k)*2^(n-k); T(2*n+1,2*k) = binomial(n,k)*2^(n-k+1).
Recurrence equations:
T(2*n,2*k) = n/k*T(2*n-1,2*k-1), T(2*n,2*k+1) = n*T(2*n-1,2*k);
T(2*n+1,2*k) = 1/k*T(2*n,2*k-1), T(2*n+1,2*k+1) = T(2*n,2*k).
O.g.f.: P(x,t)/Q(x,t), where P(x,t) = 1 + (x+2)*t - (1-x)^2*t^2 - (x^3+2*x^2+x+4)*t^3 and Q(x,t) = (1-(x^2+2)*t^2)^2.
Row polynomials:
R(2*n,x) = (x^2+2*n*x+n+2)*(x^2+2)^(n-1);
R(2*n+1,x) = (x^3+2*x^2+(n+2)*x+4)*(x^2+2)^(n-1).
Column 0 = A211227. Row sums A211231.

A211231 Row sums of A211230.

Original entry on oeis.org

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

Views

Author

Peter Bala, Apr 05 2012

Keywords

Crossrefs

Cf. A211227, A211230.

Formula

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 + ....
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.