A103353
First column of triangular matrix A103244.
Original entry on oeis.org
1, 2, 20, 512, 25392, 2093472, 260555392, 45819233280, 10849051434240, 3334632688448000, 1292876470540099584, 617862114722159788032, 357118557050589336432640, 245715466325821945360588800, 198568949299946066906578944000, 186309450278791634742517692366848
Offset: 1
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nmax = 16;
P = Table[If[n >= k, (-k^2-k)^(n-k)/(n-k)!, 0], {n, 1, nmax}, {k, 1, nmax}] // Inverse;
T[n_, k_] := If[n < k || k < 1, 0, P[[n, k]] (n - k)!];
a[n_] := T[n, 1];
Array[a, nmax] (* Jean-François Alcover, Aug 09 2018, from PARI *)
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{a(n)=local(P);if(n>=1, P=matrix(n,n,r,c,if(r>=c,(-c^2-c)^(r-c)/(r-c)!))); return(if(n<1,0,(P^-1)[n,1]*(n-1)!))}
A103238
Triangular matrix T, read by rows, that satisfies: T^2 + T = SHIFTUP(T), also T^(n+1) + T^n = SHIFTUP(T^n - D*T^(n-1)) for all n, where D is a diagonal matrix with diagonal(D) = diagonal(T) = {1,2,3,...}.
Original entry on oeis.org
1, 2, 2, 8, 6, 3, 52, 36, 12, 4, 480, 324, 96, 20, 5, 5816, 3888, 1104, 200, 30, 6, 87936, 58536, 16320, 2800, 360, 42, 7, 1601728, 1064016, 294048, 49200, 5940, 588, 56, 8, 34251520, 22728384, 6252288, 1032800, 120960, 11172, 896, 72, 9, 843099616
Offset: 0
Rows of T begin:
[1],
[2,2],
[8,6,3],
[52,36,12,4],
[480,324,96,20,5],
[5816,3888,1104,200,30,6],
[87936,58536,16320,2800,360,42,7],
[1601728,1064016,294048,49200,5940,588,56,8],...
Rows of T^2 begin:
[1],
[6,4],
[44,30,9],
[428,288,84,16],
[5336,3564,1008,180,25],...
Then T^2 + T = SHIFTUP(T):
[2],
[8,6],
[52,36,12],
[480,324,96,20],
[5816,3888,1104,200,30],...
G.f. for column 0: 1 = (1-2x) + 2*x/(1-x)*(1-2x)(1-3x) + 8*x^2/(1-x)^2*(1-2x)(1-3x)(1-4x) + 52*x^3/(1-x)^3*(1-2x)(1-3x)(1-4x)(1-5x) + ... + T(n,0)*x^n/(1-x)^n*(1-2x)(1-3x)*..*(1-(n+2)x) + ...
G.f. for column 1: 2 = 2*(1-3x) + 6*x/(1-x)*(1-3x)(1-4x) + 36*x^2/(1-x)^2*(1-3x)(1-4x)(1-5x) + 324*x^3/(1-x)^3*(1-3x)(1-4x)(1-5x)(1-6x) + ... + T(n,1)*x^(n-1)/(1-x)^(n-1)*(1-3x)(1-4x)*..*(1-(n+2)x) + ...
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/* Using Matrix Diagonalization Formula: */ T(n,k)=my(P,D);D=matrix(n+1,n+1,r,c,if(r==c,r)); P=matrix(n+1,n+1,r,c,if(r>=c,(-1)^(r-c)*(c^2+c)^(r-c)/(r-c)!)); return(if(n
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/* Using Generating Function for Columns: */ T(n,k)=if(n
A261642
Triangle, read by rows, where T(n,k) = (k^2 + k)^(n-k) for k=1..n and n>=1.
Original entry on oeis.org
1, 2, 1, 4, 6, 1, 8, 36, 12, 1, 16, 216, 144, 20, 1, 32, 1296, 1728, 400, 30, 1, 64, 7776, 20736, 8000, 900, 42, 1, 128, 46656, 248832, 160000, 27000, 1764, 56, 1, 256, 279936, 2985984, 3200000, 810000, 74088, 3136, 72, 1, 512, 1679616, 35831808, 64000000, 24300000, 3111696, 175616, 5184, 90, 1
Offset: 1
This triangle begins:
1;
2, 1;
4, 6, 1;
8, 36, 12, 1;
16, 216, 144, 20, 1;
32, 1296, 1728, 400, 30, 1;
64, 7776, 20736, 8000, 900, 42, 1;
128, 46656, 248832, 160000, 27000, 1764, 56, 1;
256, 279936, 2985984, 3200000, 810000, 74088, 3136, 72, 1;
512, 1679616, 35831808, 64000000, 24300000, 3111696, 175616, 5184, 90, 1;
1024, 10077696, 429981696, 1280000000, 729000000, 130691232, 9834496, 373248, 8100, 110, 1; ...
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{T(n, k) = (k^2 + k)^(n-k)}
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print(""))
Showing 1-3 of 3 results.
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