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 A124860 A Jacobsthal-Pascal triangle. 4
 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, 42966, 42966, 28644, 12276, 3069, 341 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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 G. C. Greubel, Rows n = 0..50 of the triangle, flattened FORMULA G.f.: 1/(1 - x*(1+y) - 2*x^2*(1+y)^2). T(n, k) = J(n+1) * C(n, k), where J(n) = A001045(n). T(n, 0) = T(n, n) = A001045(n+1). T(2*n, n) = A124862(n). Sum_{k=0..n} T(n, k) = A003683(n+1). Sum_{k=0..floor(n/2)} T(n-k, k) = A124861(n). T(n, k) = T(n-1, k-1) + T(n-1, k) + 2*T(n-2, k-2) + 4*T(n-2, k-1) + 2*T(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 From G. C. Greubel, Feb 17 2023: (Start) T(n, n-k) = T(n, k). T(n, 1) = A193449(n). Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End) 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_]:= Binomial[n, k]*(2^(n+1) -(-1)^(n+1))/3; ColumnForm[Table[jacobPascal[n, k], {n, 0, 12}, {k, 0, n}], Center] (* Alonso del Arte, Jan 16 2020 *) PROG (Magma) A124860:= func< n, k | Binomial(n, k)*(2^(n+1) - (-1)^(n+1))/3 >; [A124860(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2023 (SageMath) def A124860(n, k): return binomial(n, k)*(2^(n+1) - (-1)^(n+1))/3 flatten([[A124860(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 17 2023 CROSSREFS Cf. A001045, A003683 (row sums), A016095, A084938, A124862 (diagonal sums), A193449. Sequence in context: A351101 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 November 29 16:16 EST 2023. Contains 367445 sequences. (Running on oeis4.)