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 A305601 #24 Jun 15 2018 00:30:14
%S A305601 1,-1,0,2,-7,21,-56,125,-209,154,572,-3404,11930,-35394,99144,-274550,
%T A305601 757813,-2057867,5392764,-13383194,30829315,-63995085,112304664,
%U A305601 -133488537,-70063101,1177164984,-5212740024,17744267816,-53365305260,149452146536,-401301906464,1053638673004,-2741495192679,7122821187935,-18514807074104,48019064944442,-123571120120435,313403811733896,-778001059367703,1877334690759250,-4370190271978998
%N A305601 G.f. A(x) satisfies: [x^k] (1+x)^(n*(n+1)/2) * A(x) = 0 for k = n*(n-1)/2 + 1 through k = n*(n+1)/2 for n >= 1.
%C A305601 The diagonals in the table of coefficients of x^k in (1+x)^n * A(x) forms the rows of irregular triangle A127496.
%C A305601 Equals the inverse binomial transform of A305605.
%H A305601 Paul D. Hanna, <a href="/A305601/b305601.txt">Table of n, a(n) for n = 0..2080</a>
%F A305601 G.f. A(x) also satisfies:
%F A305601 (1) A(x) = 1/(1+x) - Sum_{n>=1} A107877(n) * (x/(1+x))^(n*(n+1)/2+1) / (1+x)^n.
%F A305601 (2) [x^(n*(n-1)/2)] (1+x)^(n*(n+1)/2) * A(x) = A107877(n) for n >= 0.
%F A305601 (3) [x^(n*(n+1)/2 + 1)] (1+x)^(n*(n+1)/2) * A(x) = -A107877(n) for n >= 0.
%F A305601 (4) [x^n] (1+x)^(2*n) * A(x) = A127497(n) = A127496(n,n) for n>=0.
%F A305601 (5) [x^k] (1+x)^(n+k) * A(x) = A127496(n,k) for k = 0..n*(n+1)/2, for n>=0.
%F A305601 (6) [x^n] (1+x)^n * A(x) = A305605(n) for n >= 0.
%e A305601 G.f. A(x) = 1 - x + 2*x^3 - 7*x^4 + 21*x^5 - 56*x^6 + 125*x^7 - 209*x^8 + 154*x^9 + 572*x^10 - 3404*x^11 + 11930*x^12 - 35394*x^13 + 99144*x^14 - 274550*x^15 + 757813*x^16 + ...
%e A305601 ILLUSTRATION OF DEFINITION.
%e A305601 The table of coefficients of x^k in (1+x)^n * A(x) begins:
%e A305601 n= 0: [1, -1, 0, 2, -7, 21, -56, 125, -209, 154, 572, -3404, ...];
%e A305601 n= 1: [1, 0, -1, 2, -5, 14, -35, 69, -84, -55, 726, -2832, ...];
%e A305601 n= 2: [1, 1, -1, 1, -3, 9, -21, 34, -15, -139, 671, -2106, ...];
%e A305601 n= 3: [1, 2, 0, 0, -2, 6, -12, 13, 19, -154, 532, -1435, ...];
%e A305601 n= 4: [1, 3, 2, 0, -2, 4, -6, 1, 32, -135, 378, -903, ...];
%e A305601 n= 5: [1, 4, 5, 2, -2, 2, -2, -5, 33, -103, 243, -525, ...];
%e A305601 n= 6: [1, 5, 9, 7, 0, 0, 0, -7, 28, -70, 140, -282, ...];
%e A305601 n= 7: [1, 6, 14, 16, 7, 0, 0, -7, 21, -42, 70, -142, ...];
%e A305601 n= 8: [1, 7, 20, 30, 23, 7, 0, -7, 14, -21, 28, -72, ...];
%e A305601 n= 9: [1, 8, 27, 50, 53, 30, 7, -7, 7, -7, 7, -44, ...];
%e A305601 n=10: [1, 9, 35, 77, 103, 83, 37, 0, 0, 0, 0, -37, ...];
%e A305601 n=11: [1, 10, 44, 112, 180, 186, 120, 37, 0, 0, 0, -37, ...];
%e A305601 n=12: [1, 11, 54, 156, 292, 366, 306, 157, 37, 0, 0, -37, ...];
%e A305601 n=13: [1, 12, 65, 210, 448, 658, 672, 463, 194, 37, 0, -37, ...];
%e A305601 n=14: [1, 13, 77, 275, 658, 1106, 1330, 1135, 657, 231, 37, -37, ...];
%e A305601 n=15: [1, 14, 90, 352, 933, 1764, 2436, 2465, 1792, 888, 268, 0, 0, 0, 0, 0, -268, ...]; ...
%e A305601 which illustrates the occurrences of zeros in the table.
%e A305601 RELATED SEQUENCES.
%e A305601 Notice that [x^(n*(n-1)/2)] (1+x)^(n*(n+1)/2) * A(x) = A107877(n), which begins:
%e A305601 [1, 1, 2, 7, 37, 268, 2496, 28612, 391189, 6230646, 113521387, ...].
%e A305601 Also, note that the coefficient of x^(n*(n-1)/2) in (1+x)^(n*(n+1)/2) * A(x) yields -A107877(n).
%e A305601 Also, observe that [x^n] (1+x)^(2*n) * A(x) = A127497(n), which begins:
%e A305601 [1, 1, 2, 7, 23, 83, 306, 1135, 4257, 16095, 61222, 233956, ...].
%e A305601 The initial terms of the diagonals in the above table forms the rows of irregular triangle A127496:
%e A305601 1;
%e A305601 1, 1;
%e A305601 1, 2, 2, 2;
%e A305601 1, 3, 5, 7, 7, 7, 7;
%e A305601 1, 4, 9, 16, 23, 30, 37, 37, 37, 37, 37;
%e A305601 1, 5, 14, 30, 53, 83, 120, 157, 194, 231, 268, 268, 268, 268, 268, 268;
%e A305601 1, 6, 20, 50, 103, 186, 306, 463, 657, 888, 1156, 1424, 1692, 1960, 2228, 2496, 2496, 2496, 2496, 2496, 2496, 2496; ...
%e A305601 in which row n equals the partial sums of the prior row with the final term repeated n more times at the end.
%o A305601 (PARI) /* Informal code to generate terms */
%o A305601 {A=[1, -1]; for(i=1, 465, A=concat(A, 0); m=floor(sqrt(2*#A-2) + 1/2); A[#A] = -polcoeff( (1+x +x*O(x^#A))^(m*(m+1)/2)*Ser(A), #A-1) ; print1(#A, ", ")); A}
%o A305601 /* Show that the definition is satisfied: */
%o A305601 for(n=1, floor(sqrt(2*#A) + 1/2), print1(n": "); for(k=n*(n-1)/2+1, n*(n+1)/2, print1(polcoeff( (1+x +x*O(x^#A))^(n*(n+1)/2)*Ser(A) , k), ", ")); print(""))
%Y A305601 Cf. A107877, A127497, A127496, A305605, A305600.
%K A305601 sign
%O A305601 0,4
%A A305601 _Paul D. Hanna_, Jun 14 2018