A136467 Triangle T, read by rows, where column 0 of T^m equals C(m*2^(n-1), n) as n=0,1,2,3,..., for all m.
1, 1, 1, 1, 4, 1, 4, 32, 16, 1, 70, 848, 576, 64, 1, 4368, 75648, 62208, 9216, 256, 1, 906192, 22313216, 21169152, 3792896, 143360, 1024, 1, 621216192, 21827627008, 23212261376, 4793434112, 223215616, 2228224, 4096, 1, 1429702652400, 71889350288384, 83889221697536, 19373156990976, 1055047811072, 13257146368, 34865152, 16384, 1, 11288510714272000, 812123016027521024, 1022353118549770240, 258404578922332160, 15923445036482560, 238096880762880, 803108552704, 549453824, 65536, 1
Offset: 0
Examples
Triangle T begins: 1; 1, 1; 1, 4, 1; 4, 32, 16, 1; 70, 848, 576, 64, 1; 4368, 75648, 62208, 9216, 256, 1; 906192, 22313216, 21169152, 3792896, 143360, 1024, 1; 621216192, 21827627008, 23212261376, 4793434112, 223215616, 2228224, 4096, 1; 1429702652400, 71889350288384, 83889221697536, 19373156990976, 1055047811072, 13257146368, 34865152, 16384, 1; 11288510714272000, 812123016027521024, 1022353118549770240, 258404578922332160, 15923445036482560, 238096880762880, 803108552704, 549453824, 65536, 1; ... Column 0 of T^m is given by: [T^m](n,0) = C(m*2^(n-1), n) for n>=0. Matrix square T^2 begins: 1; 2, 1; 6, 8, 1; 56, 128, 32, 1; 1820, 6048, 2176, 128, 1; 201376, 912128, 419328, 34816, 512, 1; 74974368, 449708544, 249300992, 26198016, 548864, 2048, 1; 94525795200, 739136655360, 477013868544, 59943682048, 1604059136, 8650752, 8192, 1; ... in which column 0 equals [T^2](n,0) = C(2^n, n) for n>=0. Matrix cube T^3 begins: 1; 3, 1; 15, 12, 1; 220, 288, 48, 1; 10626, 19696, 4800, 192, 1; 1712304, 4213376, 1333504, 76800, 768, 1; 927048304, 2927926016, 1133186048, 83992576, 1216512, 3072, 1; 1708566412608, 6784661682176, 3094826778624, 278193635328, 5216272384, 19267584, 12288, 1; ... in which column 0 equals [T^3](n,0) = C(3*2^(n-1), n) for n>=0. Matrix 4th power T^4 begins: 1; 4, 1; 28, 16, 1; 560, 512, 64, 1; 35960, 45888, 8448, 256, 1; 7624512, 12731904, 3066880, 135168, 1024, 1; 5423611200, 11434738688, 3390050304, 193953792, 2146304, 4096, 1; 13161885792000, 34243130728448, 12032434503680, 841005662208, 12133597184, 34078720, 16384, 1; ... in which column 0 equals [T^4](n,0) = C(4*2^(n-1), n) for n>=0. Matrix 5th power T^5 begins: 1; 5, 1; 45, 20, 1; 1140, 800, 80, 1; 91390, 88720, 13120, 320, 1; 24040016, 30268800, 5881600, 209920, 1280, 1; 21193254160, 33353694464, 8005555200, 372858880, 3338240, 5120, 1; 63815149590720, 122539734714368, 34967493738496, 1998561607680, 23429775360, 53084160, 20480, 1; ... in which column 0 equals [T^5](n,0) = C(5*2^(n-1), n) for n>=0.
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..495
Programs
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PARI
{T(n,k) = my(M=matrix(n+1,n+1,r,c,binomial(r*2^(c-2),c-1)),P); P=matrix(n+1,n+1,r,c,binomial((r+1)*2^(c-2),c-1));(P~*M~^-1)[n+1,k+1]} for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
Formula
Diagonals: T(n+1,n) = 4^n; T(n+2,n) = (2^(n+1) + n-1)*8^n.
T(n,k) is divisible by 2^((n-k)*k) for n>=k>=0.
Comments