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%I #5 Mar 01 2021 21:51:27
%S 1,1,1,1,6,1,1,17,17,1,1,38,154,38,1,1,79,872,872,79,1,1,160,3991,
%T 14064,3991,160,1,1,321,16791,157575,157575,16791,321,1,1,642,68312,
%U 1451486,4815630,1451486,68312,642,1,1,1283,274394,12266038,107115116,107115116,12266038,274394,1283,1
%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=1, read by rows.
%C Row sums are: {1, 2, 8, 36, 232, 1904, 22368, 349376, 7856512, 239313664, 10534962688, ...}.
%H G. C. Greubel, <a href="/A154985/b154985.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=1.
%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=1. - _G. C. Greubel_, Mar 01 2021
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 6, 1;
%e 1, 17, 17, 1;
%e 1, 38, 154, 38, 1;
%e 1, 79, 872, 872, 79, 1;
%e 1, 160, 3991, 14064, 3991, 160, 1;
%e 1, 321, 16791, 157575, 157575, 16791, 321, 1;
%e 1, 642, 68312, 1451486, 4815630, 1451486, 68312, 642, 1;
%e 1, 1283, 274394, 12266038, 107115116, 107115116, 12266038, 274394, 1283, 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,1]], 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, 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 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,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 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,1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 01 2021
%Y Cf. A154983 (m=0), this sequence (m=1), A154984 (m=2).
%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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