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A154979 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=2, read by rows. 7

%I

%S 1,1,1,1,10,1,1,27,27,1,1,60,374,60,1,1,125,2162,2162,125,1,1,254,

%T 9967,52196,9967,254,1,1,511,42221,615635,615635,42221,511,1,1,1024,

%U 172780,5760960,27955622,5760960,172780,1024,1,1,2049,697068,49168044,664126822,664126822,49168044,697068,2049,1

%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=2, read by rows.

%C Row sums are: {1, 2, 12, 56, 496, 4576, 72640, 1316736, 39825152, 1427987968, 84417887232, ...}.

%H G. C. Greubel, <a href="/A154979/b154979.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=2.

%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=2. - _G. C. Greubel_, Mar 01 2021

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 10, 1;

%e 1, 27, 27, 1;

%e 1, 60, 374, 60, 1;

%e 1, 125, 2162, 2162, 125, 1;

%e 1, 254, 9967, 52196, 9967, 254, 1;

%e 1, 511, 42221, 615635, 615635, 42221, 511, 1;

%e 1, 1024, 172780, 5760960, 27955622, 5760960, 172780, 1024, 1;

%e 1, 2049, 697068, 49168044, 664126822, 664126822, 49168044, 697068, 2049, 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,2]], 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,2], {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,2) 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,2): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 01 2021

%Y Cf. A154982 (m=0), A154980 (m=1), this sequence (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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Last modified May 29 07:49 EDT 2022. Contains 354122 sequences. (Running on oeis4.)