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A124860 A Jacobsthal-Pascal triangle. 3
1, 1, 1, 3, 6, 3, 5, 15, 15, 5, 11, 44, 66, 44, 11, 21, 105, 210, 210, 105, 21, 43, 258, 645, 860, 645, 258, 43, 85, 595, 1785, 2975, 2975, 1785, 595, 85, 171, 1368, 4788, 9576, 11970, 9576, 4788, 1368, 171, 341, 3069, 12276, 28644 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row sums are A003683(n+1). Diagonal sums are A124861. Central coefficients are A124862.

Triangle T(n, k) read by rows given by [1, 2, -2, 0, 0, 0, ...] DELTA [1, 2, -2, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 11 2006

LINKS

Table of n, a(n) for n=0..48.

FORMULA

G.f.: 1/(1 - x(1 + y) - 2x^2(1 + y)^2).

T(n, k) = J(n + 1) * C(n, k), where J(n) = A001045(n).

T(n, k) = T(n - 1, k - 1) + T(n - 1, k) + 2T(n - 2, k - 2) + 4T(n - 2, k - 1) + 2T(n - 2, k), T(0, 0) = 1, T(n, k) = 0 if k < 0 or if k > n . - Philippe Deléham, Nov 11 2006

G.f.: T(0)/2, where T(k) = 1 + 1/(1 - (2*k + 1 + 2*x*(1 + y))*x*(1 + y)/((2*k + 2 + 2*x*(1 + y))*x*(1 + y)+ 1/T(k + 1))); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013

EXAMPLE

Triangle begins

   1;

   1,   1;

   3,   6,   3;

   5,  15,  15,   5;

  11,  44,  66,  44,  11;

  21, 105, 210, 210, 105,  21;

  43, 258, 645, 860, 645, 258, 43;

MAPLE

A := proc(n, k) ## n >= 0 and k = 0 .. n

    ((-1)^n+2^(n+1))/3*binomial(n, k)

end proc: # Yu-Sheng Chang, Jan 15 2020

MATHEMATICA

jacobPascal[n_, k_] := (2^(n + 1) - (-1)^(n + 1))/3 Binomial[n, k]; ColumnForm[Table[jacobPascal[n, k], {n, 0, 9}, {k, 0, n}], Center] (* Alonso del Arte, Jan 16 2020 *)

CROSSREFS

Cf. A001045 (column 0, column k), A016095.

Sequence in context: A199951 A134548 A151865 * A182412 A038138 A010704

Adjacent sequences:  A124857 A124858 A124859 * A124861 A124862 A124863

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Nov 10 2006

STATUS

approved

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Last modified July 12 18:03 EDT 2020. Contains 335666 sequences. (Running on oeis4.)