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A257612
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Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.
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10
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1, 2, 2, 4, 24, 4, 8, 184, 184, 8, 16, 1216, 3680, 1216, 16, 32, 7584, 53824, 53824, 7584, 32, 64, 46208, 674752, 1507072, 674752, 46208, 64, 128, 278912, 7764096, 33244544, 33244544, 7764096, 278912, 128, 256, 1677312, 84892672, 636233728, 1196803584, 636233728, 84892672, 1677312, 256
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OFFSET
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0,2
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COMMENTS
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Corresponding entries in this triangle and in A060187 differ only by powers of 2. - F. Chapoton, Nov 04 2020
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LINKS
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FORMULA
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T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 4, and b = 2. - G. C. Greubel, Mar 20 2022
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EXAMPLE
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Triangle begins as:
1;
2, 2;
4, 24, 4;
8, 184, 184, 8;
16, 1216, 3680, 1216, 16;
32, 7584, 53824, 53824, 7584, 32;
64, 46208, 674752, 1507072, 674752, 46208, 64;
128, 278912, 7764096, 33244544, 33244544, 7764096, 278912, 128;
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MATHEMATICA
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T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n, k, 4, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)
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PROG
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(PARI) f(x) = 4*x + 2;
T(n, k) = t(n-k, k);
t(n, m) = if (!n && !m, 1, if (n < 0 || m < 0, 0, f(m)*t(n-1, m) + f(n)*t(n, m-1)));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", "); ); print(); ); \\ Michel Marcus, May 06 2015
(Sage)
if (k<0 or k>n): return 0
elif (n==0): return 1
else: return (a*k+b)*T(n-1, k, a, b) + (a*(n-k)+b)*T(n-1, k-1, a, b)
flatten([[T(n, k, 4, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022
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CROSSREFS
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See similar sequences listed in A256890.
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KEYWORD
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AUTHOR
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STATUS
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approved
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