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 A304191 #25 Nov 10 2019 20:26:08
%S A304191 1,1,3,35,611,14691,448873,16606825,720241161,35786093321,
%T A304191 2002505540123,124546575282555,8520012343770331,635618668572015451,
%U A304191 51348334729127568273,4465119223213849398545,415808496978034659793361,41283870149540066960271441,4353184675864365012327673843,485828603554439779231472806675
%N A304191 G.f. A(x) satisfies: [x^n] (1+x)^(n^2) / A(x) = 0 for n > 0.
%C A304191 Note that [x^n] (1+x)^(n*k) / G(x) = 0 for n > 0 holds when G(x) = (1+x)/(1 - (k-1)*x) given some fixed k; this sequence explores the case where k varies with n.
%H A304191 Paul D. Hanna, <a href="/A304191/b304191.txt">Table of n, a(n) for n = 0..300</a>
%F A304191 A132617(n+1) = [x^n] (1+x)^((n+1)^2) / A(x) for n >= 0.
%e A304191 G.f.: A(x) = 1 + x + 3*x^2 + 35*x^3 + 611*x^4 + 14691*x^5 + 448873*x^6 + 16606825*x^7 + 720241161*x^8 + 35786093321*x^9 + 2002505540123*x^10 + ...
%e A304191 ILLUSTRATION OF DEFINITION.
%e A304191 (EX. 1) The table of coefficients of x^k in (1+x)^(n^2) / A(x) begins:
%e A304191 n=0: [1, -1, -2, -30, -540, -13380, -416910, -15634290, ...];
%e A304191 n=1: [1, 0, -3, -32, -570, -13920, -430290, -16051200, ...];
%e A304191 n=2: [1, 3, 0, -40, -675, -15729, -473792, -17384400, ...];
%e A304191 n=3: [1, 8, 25, 0, -840, -19488, -559584, -19917600, ...];
%e A304191 n=4: [1, 15, 102, 378, 0, -24192, -712590, -24272754, ...];
%e A304191 n=5: [1, 24, 273, 1920, 8460, 0, -883740, -31495200, ...];
%e A304191 n=6: [1, 35, 592, 6408, 48885, 252087, 0, -39049296, ...];
%e A304191 n=7: [1, 48, 1125, 17120, 189090, 1583040, 9392890, 0, ...]; ...
%e A304191 in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^(n^2) / A(x) = 0 for n > 0.
%e A304191 RELATED SEQUENCES.
%e A304191 (EX. 2) The secondary diagonal in the above table (EX. 1) that begins
%e A304191 [1, 3, 25, 378, 8460, 252087, 9392890, 420142350, ...]
%e A304191 yields A132617, column 1 of triangle A132615.
%e A304191 Related triangular matrix T = A132615 begins:
%e A304191 1;
%e A304191 1, 1;
%e A304191 1, 1, 1;
%e A304191 6, 3, 1, 1;
%e A304191 80, 25, 5, 1, 1;
%e A304191 1666, 378, 56, 7, 1, 1;
%e A304191 47232, 8460, 1020, 99, 9, 1, 1;
%e A304191 1694704, 252087, 26015, 2134, 154, 11, 1, 1;
%e A304191 73552752, 9392890, 855478, 61919, 3848, 221, 13, 1, 1; ...
%e A304191 in which row n equals row (n-1) of T^(2*n-1) followed by '1' for n > 0.
%e A304191 (EX. 3) The next diagonal in the table (EX. 1) that begins:
%e A304191 [1, 8, 102, 1920, 48885, 1583040, 1583040, 62467314, ...]
%e A304191 yields the first column in the following matrix product.
%e A304191 Let TSL(m) denote the table T = A132615, with the diagonal of 1's truncated, as SHIFTED LEFT m times, so that
%e A304191 TSL(1) begins
%e A304191 [ 1];
%e A304191 [ 3, 1];
%e A304191 [ 25, 5, 1];
%e A304191 [ 378, 56, 7, 1];
%e A304191 [8460, 1020, 99, 9, 1]; ...
%e A304191 TSL(2) begins
%e A304191 [ 1];
%e A304191 [ 5, 1];
%e A304191 [ 56, 7, 1];
%e A304191 [ 1020, 99, 9, 1];
%e A304191 [26015, 2134, 154, 11, 1]; ...
%e A304191 etc.,
%e A304191 then the matrix product TSL(2)*TSL(1) begins
%e A304191 [ 1];
%e A304191 [ 8, 1];
%e A304191 [ 102, 12, 1];
%e A304191 [ 1920, 200, 16, 1];
%e A304191 [ 48885, 4540, 330, 20, 1];
%e A304191 [ 1583040, 132810, 8816, 492, 24, 1];
%e A304191 [62467314, 4790156, 293419, 15148, 686, 28, 1]; ...
%e A304191 in which the first column equals the secondary diagonal in the table of (EX. 1).
%e A304191 The subsequent diagonal in the table of (EX. 1) also equals the first column of matrix product TSL(3)*TSL(2)*TSL(1). This process can be continued to produce all the lower diagonals of the table of (EX. 1).
%o A304191 (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^((m-1)^2)/Ser(A) )[m] ); A[n+1]}
%o A304191 for(n=0,30, print1(a(n),", "))
%Y A304191 Cf. A132617, A304190, A304192, A304193, A132615.
%K A304191 nonn
%O A304191 0,3
%A A304191 _Paul D. Hanna_, May 07 2018