This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A102765 #10 Feb 13 2022 09:25:56 %S A102765 0,0,1,0,1,3,0,1,5,13,0,1,7,25,51,0,1,9,43,125,205,0,1,11,67,259,625, %T A102765 819,0,1,13,97,477,1555,3125,3277,0,1,15,133,803,3355,9331,15625, %U A102765 13107,0,1,17,175,1261,6505,23517,55987,78125,52429,0,1,19,223,1875,11605 %N A102765 Array read by antidiagonals: T(n, k) = ((n+4)^k-(n-1)^k)/5. %C A102765 Consider a 5x5 matrix M = %C A102765 [n, 1, 1, 1, 1] %C A102765 [1, n, 1, 1, 1] %C A102765 [1, 1, n, 1, 1] %C A102765 [1, 1, 1, n, 1] %C A102765 [1, 1, 1, 1, n]. %C A102765 The n-th row of the array contains the values of the non diagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = non diagonal entry + (n-1)^k.) %C A102765 For row r we have polynomial ((r+4)^n-(r-1)^n)/5. Corresponding g.f.s: x/((1-(r-1)x)(1-(r+4)x)) %C A102765 If r(n) denotes a row sequence, r(n+1)/r(n) converges to n+4. %C A102765 Triangle T(n, k) = (4^(n-k-1)-(-1)^(n-k-1))/5*(binomial(k+(n-k-1),n-k-1)) gives coefficients for polynomials for the columns of the array. First four polynomial are: %C A102765 1 %C A102765 3 + 2*k %C A102765 13 + 9*k + 3*k^2 %C A102765 51 + 52*k + 18*k^2 + 4*k^3 %C A102765 ... %e A102765 Array begins: %e A102765 0, 1, 3, 13, 51, 205, ... %e A102765 0, 1, 5, 25, 125, 625, ... %e A102765 0, 1, 7, 43, 259, 1555, ... %e A102765 0, 1, 9, 67, 477, 3355, ... %e A102765 0, 1, 11, 97, 803, 6505, ... %e A102765 ... %o A102765 (PARI) MM(n,N)=local(M);M=matrix(n,n);for(i=1,n, for(j=1,n,if(i==j,M[i,j]=N,M[i,j]=1)));M %o A102765 for(k=0,10, for(i=0,10,print1((MM(5,k)^i)[1,2],","));print()) %o A102765 (PARI) p(n,k)=((n+4)^k-(n-1)^k)/5 %o A102765 for(k=0,10, for(i=0,10,print1(p(k,i),","));print()) %o A102765 (PARI) for(k=0,10, for(i=0,10,print1(polcoeff(x/((1-(k-1)*x)*(1-(k+4)*x)),i),","));print()) %Y A102765 Cf. A015521 (for n=0), A000351 (for n=1), A003464 (for n=2), A016130 (for n=3), A016140 (for n=4), A016153 (for n=5), A016164 (for n=6), A016174 (for n=7), A016184 (for n=8), A015441 (for n=-1), A091005 (for n=-2). %K A102765 nonn,tabl %O A102765 0,6 %A A102765 Lambert Klasen (lambert.klasen(AT)gmx.net) and _Gary W. Adamson_, Feb 10 2005