A257613 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) = 2*x + 4.
1, 4, 4, 16, 48, 16, 64, 416, 416, 64, 256, 3136, 6656, 3136, 256, 1024, 21888, 84608, 84608, 21888, 1024, 4096, 145664, 939520, 1692160, 939520, 145664, 4096, 16384, 939520, 9555456, 28195840, 28195840, 9555456, 939520, 16384, 65536, 5932032, 91475968, 415734784, 676700160, 415734784, 91475968, 5932032, 65536
Offset: 0
Examples
Triangle begins as:
1;
4, 4;
16, 48, 16;
64, 416, 416, 64;
256, 3136, 6656, 3136, 256;
1024, 21888, 84608, 84608, 21888, 1024;
4096, 145664, 939520, 1692160, 939520, 145664, 4096;
16384, 939520, 9555456, 28195840, 28195840, 9555456, 939520, 16384;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Crossrefs
Programs
-
Mathematica
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,2,4], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *) -
PARI
f(x) = 2*x + 4; 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
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Sage
def T(n,k,a,b): # A257613 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,2,4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022
Formula
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) = 2*x + 4.
Sum_{k=0..n} T(n, k) = A051580(n).
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 = 2, and b = 4. - G. C. Greubel, Mar 20 2022