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%I #5 Mar 01 2021 17:53:32
%S 1,1,1,1,6,1,1,15,15,1,1,32,126,32,1,1,65,638,638,65,1,1,130,2751,
%T 9340,2751,130,1,1,259,11201,93755,93755,11201,259,1,1,516,44740,
%U 809212,2578550,809212,44740,516,1,1,1029,177864,6588864,51390322,51390322,6588864,177864,1029,1
%N Triangle T(n, k, m) = coefficients of p(x, n, m) where p(x,n,m) = (x+1)*p(x, n-1, m) + 2^(m+n-1) *x*p(x, n-2, m) and m=1, read by rows.
%C Row sums are: {1, 2, 8, 32, 192, 1408, 15104, 210432, 4287488, 116316160, 4623020032, ...}.
%H G. C. Greubel, <a href="/A154980/b154980.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k, m) = coefficients of p(x, n, m) where p(x,n,m) = (x+1)*p(x, n-1, m) + 2^(m+n-1) *x*p(x, n-2, m) and m=1.
%F T(n, k, m) = T(n-1, k, m) + T(n-1, k-1, m) + 2^(n+m-1)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m=1. - _G. C. Greubel_, Mar 01 2021
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 6, 1;
%e 1, 15, 15, 1;
%e 1, 32, 126, 32, 1;
%e 1, 65, 638, 638, 65, 1;
%e 1, 130, 2751, 9340, 2751, 130, 1;
%e 1, 259, 11201, 93755, 93755, 11201, 259, 1;
%e 1, 516, 44740, 809212, 2578550, 809212, 44740, 516, 1;
%e 1, 1029, 177864, 6588864, 51390322, 51390322, 6588864, 177864, 1029, 1;
%t (* First program *)
%t p[x_, n_, m_]:= p[x,n,m] = If[n<2, n*x+1, (x+1)*p[x,n-1,m] + 2^(m+n-1)*x*p[x, n-2, m]];
%t Table[CoefficientList[ExpandAll[p[x,n,1]], x], {n,0,12}]//Flatten (* modified by _G. C. Greubel_, Mar 01 2021 *)
%t (* Second program *)
%t T[n_, k_, m_]:= T[n,k,m] = If[k==0 || k==n, 1, T[n-1, k, m] + T[n-1, k-1, m] + 2^(n+m-1)*T[n-2, k-1, m]];
%t Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 01 2021 *)
%o (Sage)
%o def T(n,k,m):
%o if (k==0 or k==n): return 1
%o else: return T(n-1, k, m) + T(n-1, k-1, m) + 2^(n+m-1)*T(n-2, k-1, m)
%o flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 01 2021
%o (Magma)
%o function T(n,k,m)
%o if k eq 0 or k eq n then return 1;
%o else return T(n-1, k, m) + T(n-1, k-1, m) + 2^(n+m-1)*T(n-2, k-1, m);
%o end if; return T;
%o end function;
%o [T(n,k,1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 01 2021
%Y Cf. A154982 (m=0), this sequence (m=1), A154979 (m=3).
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Jan 18 2009
%E Edited by _G. C. Greubel_, Mar 01 2021
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