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%I #5 Mar 01 2021 21:51:07
%S 1,1,1,1,4,1,1,11,11,1,1,24,70,24,1,1,49,358,358,49,1,1,98,1559,4076,
%T 1559,98,1,1,195,6361,40003,40003,6361,195,1,1,388,25372,345692,
%U 862598,345692,25372,388,1,1,773,100640,2813688,16569442,16569442,2813688,100640,773,1
%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) + 2^(n-2)*[n>=3])*x*p(x, n-2, m) and m=0, read by rows.
%C Row sums are: {1, 2, 6, 24, 120, 816, 7392, 93120, 1605504, 38969088, 1310965248, ...}.
%H G. C. Greubel, <a href="/A154983/b154983.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) + 2^(n-2)*[n>=3])*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) + 2^(n-2)*[n>=3])*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, 11, 11, 1;
%e 1, 24, 70, 24, 1;
%e 1, 49, 358, 358, 49, 1;
%e 1, 98, 1559, 4076, 1559, 98, 1;
%e 1, 195, 6361, 40003, 40003, 6361, 195, 1;
%e 1, 388, 25372, 345692, 862598, 345692, 25372, 388, 1;
%e 1, 773, 100640, 2813688, 16569442, 16569442, 2813688, 100640, 773, 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] + Boole[n>=3]*2^(n-2)*x*p[x, n-2, m] ];
%t Table[CoefficientList[ExpandAll[p[x,n,0]], x], {n,0,10}]//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^(m+n-1) + Boole[n>=3]*2^(n-2))*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 elif (n<3): return T(n-1, k, m) + T(n-1, k-1, m) + 2^(n+m-1)*T(n-2, k-1, m)
%o else: return T(n-1, k, m) + T(n-1, k-1, m) + (2^(n+m-1) +2^(n-2))*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 elif (n lt 3) then return T(n-1, k, m) + T(n-1, k-1, m) + 2^(n+m-1)*T(n-2, k-1, m);
%o else return T(n-1, k, m) + T(n-1, k-1, m) + (2^(n+m-1)+2^(n-2))*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), A154984 (m=1), A154985 (m=3).
%Y Cf. A154979, A154980, A154982, A154986.
%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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