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 A126791 Binomial matrix applied to A111418. 24
 1, 4, 1, 17, 7, 1, 75, 39, 10, 1, 339, 202, 70, 13, 1, 1558, 1015, 425, 110, 16, 1, 7247, 5028, 2400, 771, 159, 19, 1, 34016, 24731, 12999, 4872, 1267, 217, 22, 1, 160795, 121208, 68600, 28882, 8890, 1940, 284, 25, 1, 764388, 593019, 355890, 164136 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Triangle T(n,k), 0<=k<=n, read by rows defined by : T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=4*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+3*T(n-1,k)+T(n-1,k+1) for k>=1. This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; ((1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906 . - Philippe Deléham, Sep 25 2007 The matrix inverse starts 1; -4,1; 11,-7,1; -29,31,-10,1; 76,-115,60,-13,1; -199,390,-285,98,-16,1; 521,-1254,1185,-566,145,-19,1; -1364,3893,-4524,2785,-985,201,-22,1; ... - R. J. Mathar, Mar 12 2013 LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened FORMULA Sum{k, k>=0}T(m,k)*T(n,k) = T(m+n,0) = A026378(m+n+1). Sum{k, 0<=k<=n}T(n,k) = 5^n = A000351(n). T(n,k) = (-1)^(n-k)*(GegenbauerC(n-k,-n+1,3/2) - GegenbauerC(n-k-1,-n+1,3/2)). - Peter Luschny, May 13 2016 EXAMPLE Triangle begins: 1; 4, 1; 17, 7, 1; 75, 39, 10, 1; 339, 202, 70, 13, 1; 1558, 1015, 425, 110, 16, 1; 7247, 5028, 2400, 771, 159, 19, 1; 34016, 24731, 12999, 4872, 1267, 217, 22, 1;... Production matrix begins 4, 1 1, 3, 1 0, 1, 3, 1 0, 0, 1, 3, 1 0, 0, 0, 1, 3, 1 0, 0, 0, 0, 1, 3, 1 0, 0, 0, 0, 0, 1, 3, 1 0, 0, 0, 0, 0, 0, 1, 3, 1 0, 0, 0, 0, 0, 0, 0, 1, 3, 1 - From Philippe Deléham, Nov 07 2011 MAPLE A126791 := proc(n, k)     if n=0 and k = 0 then         1 ;     elif k <0 or k>n then         0;     elif k= 0 then         4*procname(n-1, 0)+procname(n-1, 1) ;     else         procname(n-1, k-1)+3*procname(n-1, k)+procname(n-1, k+1) ;     end if; end proc: # R. J. Mathar, Mar 12 2013 T := (n, k) -> (-1)^(n-k)*simplify(GegenbauerC(n-k, -n+1, 3/2) - GegenbauerC(n-k-1, -n+1, 3/2)): seq(seq(T(n, k), k=1..n), n=1..10); # Peter Luschny, May 13 2016 MATHEMATICA T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0, T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]]; Table[T[n, k, 4, 3], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *) CROSSREFS Sequence in context: A209411 A093035 A301624 * A052179 A171589 A126331 Adjacent sequences:  A126788 A126789 A126790 * A126792 A126793 A126794 KEYWORD nonn,tabl AUTHOR Philippe Deléham, Mar 14 2007 STATUS approved

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Last modified October 15 19:25 EDT 2019. Contains 328037 sequences. (Running on oeis4.)