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A154984
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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=2, read by rows.
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4
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1, 1, 1, 1, 10, 1, 1, 29, 29, 1, 1, 66, 418, 66, 1, 1, 139, 2572, 2572, 139, 1, 1, 284, 12215, 65336, 12215, 284, 1, 1, 573, 52531, 818287, 818287, 52531, 573, 1, 1, 1150, 216688, 7906658, 39270110, 7906658, 216688, 1150, 1, 1, 2303, 877934, 68639058, 989843392, 989843392, 68639058, 877934, 2303, 1
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OFFSET
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0,5
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COMMENTS
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Row sums are: {1, 2, 12, 60, 552, 5424, 90336, 1742784, 55519104, 2118725376, 132153466368, ...}.
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LINKS
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FORMULA
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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=2.
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=2. - G. C. Greubel, Mar 01 2021
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EXAMPLE
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Triangle begins as:
1;
1, 1;
1, 10, 1;
1, 29, 29, 1;
1, 66, 418, 66, 1;
1, 139, 2572, 2572, 139, 1;
1, 284, 12215, 65336, 12215, 284, 1;
1, 573, 52531, 818287, 818287, 52531, 573, 1;
1, 1150, 216688, 7906658, 39270110, 7906658, 216688, 1150, 1;
1, 2303, 877934, 68639058, 989843392, 989843392, 68639058, 877934, 2303, 1;
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MATHEMATICA
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(* First program *)
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] ];
Table[CoefficientList[ExpandAll[p[x, n, 2]], x], {n, 0, 10}]//Flatten (* modified by G. C. Greubel, Mar 01 2021 *)
(* Second program *)
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] ];
Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 01 2021 *)
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PROG
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(Sage)
def T(n, k, m):
if (k==0 or k==n): return 1
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)
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)
flatten([[T(n, k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 01 2021
(Magma)
function T(n, k, m)
if k eq 0 or k eq n then return 1;
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);
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);
end if; return T;
end function;
[T(n, k, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 01 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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