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%I #5 Mar 01 2021 17:53:41
%S 1,1,1,1,4,1,1,9,9,1,1,18,50,18,1,1,35,212,212,35,1,1,68,823,2024,823,
%T 68,1,1,133,3131,16415,16415,3131,133,1,1,262,11968,124890,291902,
%U 124890,11968,262,1,1,519,46278,938394,4619032,4619032,938394,46278,519,1
%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=0, read by rows.
%C Row sums are: {1, 2, 6, 20, 88, 496, 3808, 39360, 566144, 11208448, ...}.
%H G. C. Greubel, <a href="/A154982/b154982.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=0.
%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=0. - _G. C. Greubel_, Mar 01 2021
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 9, 9, 1;
%e 1, 18, 50, 18, 1;
%e 1, 35, 212, 212, 35, 1;
%e 1, 68, 823, 2024, 823, 68, 1;
%e 1, 133, 3131, 16415, 16415, 3131, 133, 1;
%e 1, 262, 11968, 124890, 291902, 124890, 11968, 262, 1;
%e 1, 519, 46278, 938394, 4619032, 4619032, 938394, 46278, 519, 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,0]], 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,0], {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,0) 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,0): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 01 2021
%Y Cf. this sequence (m=0), A154980 (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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