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Equals ConvOffsStoT transform of the 2^n series: (1, 2, 4, 8, ...); e.g., ConvOffs transform of (1, 2, 4, 8) = (1, 8, 16, 8, 1). - Gary W. Adamson, Apr 21 2008

(MAGMAMagma)

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Triangle, T(n,k) = 2^(k*(n-k)), read by rows, defined by: T(n,k) = 2^((n-k)*k) for n>=k>=0.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 16, 8, 1, 1, 16, 64, 64, 16, 1, 1, 32, 256, 512, 256, 32, 1, 1, 64, 1024, 4096, 4096, 1024, 64, 1, 1, 128, 4096, 32768, 65536, 32768, 4096, 128, 1, 1, 256, 16384, 262144, 1048576, 1048576, 262144, 16384, 256, 1, 1, 512, 65536

G. C. Greubel, <a href="/A117401/b117401.txt">Rows n = 0..50 of the triangle, flattened</a>

T(n,k) = (1/n)*[( 2^(n-k)*k*T(n-1,k-1) + 2^k*(n-k)*T(n-1,k)], ), where T(i,j)=0 if j>i. - Tom Edgar, Feb 20 2014

T(n, k, m) = (m+2)^(k*(n-k)) with m = 0. - G. C. Greubel, Jun 28 2021

1;

1, 1;

1, 2, 1;

1, 4, 4, 1;

1, 8, 16, 8, 1;

1, 16, 64, 64, 16, 1;

1, 32, 256, 512, 256, 32, 1;

1, 64, 1024, 4096, 4096, 1024, 64, 1;

1, 128, 4096, 32768, 65536, 32768, 4096, 128, 1;

1, 256, 16384, 262144, 1048576, 1048576, 262144, 16384, 256, 1; ...

(MAGMA)

A117401:= func< n, k, m | (m+2)^(k*(n-k)) >;

[A117401(n, k, 0): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 28 2021

(Sage)

def A117401(n, k, m): return (m+2)^(k*(n-k))

flatten([[A117401(n, k, 0) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 28 2021

Cf. this sequence (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15).

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Let E(x) = Sum_{n>=0 } x^n/2^C(n,2). Then E(x)*E(y*x) = Sum_{n>=0 } Sum_{k=0..n} T(n,k)*y^k*x^n/2^C(n,2). - Geoffrey Critzer, May 31 2020

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