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A257606
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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) = x + 4.
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6
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1, 4, 4, 16, 40, 16, 64, 296, 296, 64, 256, 1928, 3552, 1928, 256, 1024, 11688, 34808, 34808, 11688, 1024, 4096, 67656, 302352, 487312, 302352, 67656, 4096, 16384, 379240, 2423016, 5830000, 5830000, 2423016, 379240, 16384, 65536, 2076424, 18330496, 62617144, 93280000, 62617144, 18330496, 2076424, 65536
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refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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LINKS
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FORMULA
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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) = x + 4.
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 = 1, and b = 4.
T(n, n-k) = T(n, k).
T(n, 1) = 8*5^n - 4^n*(8+n).
T(n, 2) = 2*((56 +15*n +n^2)*4^(n-1) - 4*(8+n)*5^n + 3*6^(n+1)). (End)
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EXAMPLE
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Triangle begins as:
1;
4, 4;
16, 40, 16;
64, 296, 296, 64;
256, 1928, 3552, 1928, 256;
1024, 11688, 34808, 34808, 11688, 1024;
4096, 67656, 302352, 487312, 302352, 67656, 4096;
16384, 379240, 2423016, 5830000, 5830000, 2423016, 379240, 16384;
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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, 1, 4], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)
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PROG
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(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, 1, 4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022
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CROSSREFS
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Similar sequences listed in A256890.
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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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