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A114202 A Pascal-Jacobsthal triangle. 4
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 8, 4, 1, 1, 5, 16, 16, 5, 1, 1, 6, 27, 42, 27, 6, 1, 1, 7, 41, 87, 87, 41, 7, 1, 1, 8, 58, 156, 216, 156, 58, 8, 1, 1, 9, 78, 254, 456, 456, 254, 78, 9, 1, 1, 10, 101, 386, 860, 1122, 860, 386, 101, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are A114203. T(2n,n) is A114204. Inverse has row sums 0^n.

LINKS

Table of n, a(n) for n=0..65.

Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

FORMULA

As a number triangle, with J(n) = A001045(n):

T(n, k) = Sum_{i=0..n-k} C(n-k, i)*C(k, i)*J(i);

T(n, k) = Sum_{i=0..n} C(n-k, n-i)*C(k, i-k)*J(i-k);

T(n, k) = Sum_{i=0..n} C(k, i)*C(n-k, n-i)*J(k-i) if k <= n, and 0 otherwise.

As a square array, with J(n) = A001045(n):

T(n, k) = Sum_{i=0..n} C(n, i)C(k, i)*J(i);

T(n, k) = Sum_{i=0..n+k} C(n, n+k-i)*C(k, i-k)*J(i-k);

Column k has g.f. (Sum_{i=0..k} C(k, i)*J(i+1)*(x/(1 - x))^i)*x^k/(1 - x).

EXAMPLE

Triangle begins

  1;

  1, 1;

  1, 2,  1;

  1, 3,  3,  1;

  1, 4,  8,  4,  1;

  1, 5, 16, 16,  5,  1;

  1, 6, 27, 42, 27,  6, 1;

  1, 7, 41, 87, 87, 41, 7, 1;

  ...

CROSSREFS

Sequence in context: A073134 A300260 A026692 * A125806 A202756 A156354

Adjacent sequences:  A114199 A114200 A114201 * A114203 A114204 A114205

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Nov 16 2005

STATUS

approved

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Last modified April 21 10:45 EDT 2019. Contains 322328 sequences. (Running on oeis4.)