A233531 G.f. A(x) such that triangle A233530, which transforms diagonals in the table of successive iterations of A(x), consists of all zeros after row 1.

1, 1, -2, 6, -18, 44, -56, -300, 2024, -22022, -130456, -4241064, -103538532, -2893308780, -88314189664, -2924814872208, -104538530634844, -4010605941377292, -164409679858874856, -7172735079437282200, -331847552362286195156, -16229743737669369558956, -836695536495554388520400

Offset: 1

Author

Paul D. Hanna, Dec 11 2013

Keywords

• sign

Examples

			G.f.: A(x) = x + x^2 - 2*x^3 + 6*x^4 - 18*x^5 + 44*x^6 - 56*x^7 - 300*x^8 + 2024*x^9 - 22022*x^10 - 130456*x^11 - 4241064*x^12 - 103538532*x^13 - 2893308780*x^14 - 88314189664*x^15 - 2924814872208*x^16 +...
If we form a table of coefficients in the iterations of A(x) like so:
[1,  0,   0,   0,    0,     0,      0,      0,       0,        0, ...];
[1,  1,  -2,   6,  -18,    44,    -56,   -300,    2024,   -22022, ...];
[1,  2,  -2,   3,    2,   -48,    228,   -734,   -1298,   -14630, ...];
[1,  3,   0,  -3,   18,   -54,    -24,    625,   -6324,   -46064, ...];
[1,  4,   4,  -6,   12,    26,   -332,    244,   -2078,  -108754, ...];
[1,  5,  10,   0,  -10,    90,   -192,  -2044,   -3190,  -137176, ...];
[1,  6,  18,  21,  -18,    54,    312,  -3178,  -22032,  -203692, ...];
[1,  7,  28,  63,   42,   -28,    616,   -931,  -46722,  -457746, ...];
[1,  8,  40, 132,  248,   156,    504,   3144,  -51348,  -913356, ...];
[1,  9,  54, 234,  702,  1296,   1656,   6924,  -24444, -1366530, ...];
[1, 10,  70, 375, 1530,  4580,   9916,  22122,   38570, -1538042, ...];
[1, 11,  88, 561, 2882, 11814,  38280, 104929,  273592,  -987932, ...];
[1, 12, 108, 798, 4932, 25542, 110604, 407932, 1351614,  2563858, ...]; ...
then the triangle A233530, that transforms one diagonal in the above table into another, consists of all zeros in column 0 after row 1:
1;
1, 1;
0, 2, 1;
0, 3, 3, 1;
0, 8, 9, 4, 1;
0, 38, 40, 18, 5, 1;
0, 268, 264, 112, 30, 6, 1;
0, 2578, 2379, 953, 240, 45, 7, 1;
0, 31672, 27568, 10500, 2505, 440, 63, 8, 1;
0, 475120, 392895, 143308, 32686, 5445, 728, 84, 9, 1;
0, 8427696, 6663624, 2342284, 514660, 82176, 10423, 1120, 108, 10, 1;
0, 172607454, 131211423, 44677494, 9514570, 1467837, 178689, 18214, 1632, 135, 11, 1; ...
Illustrate how T=A233530 transforms one diagonal in the above table into another:
T*[1, 1, -2, -3, 12, 90, 312, -931, -51348, -1366530, ...]
= [1, 2, 0, -6, -10, 54, 616, 3144, -24444, -1538042, ...];
T*[1, 2, 0, -6, -10, 54, 616, 3144, -24444, -1538042, ...]
= [1, 3, 4,  0, -18,-28, 504, 6924,  38570,  -987932, ...];
T*[1, 3, 4,  0, -18,-28, 504, 6924,  38570,  -987932, ...]
= [1, 4,10, 21,  42,156,1656,22122, 273592,  2563858, ...].
		

Crossrefs

Cf. A233530.

Programs

PARI

/* Given A = g.f. A(x), Calculate Triangle A233530: */
{T(n, k)=local(F=x, M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x;
for(i=1, r+c-2, F=subst(F, x, A +x*O(x^(m+2)))); polcoeff(F, c));
N=matrix(m+1, m+1, r, c, M[r, c]);
P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
/* Calculates A = g.f. A(x) and then Prints ROWS of Triangle: */
{ROWS=20;V=[1,1];print("");print1("This Sequence: [1, 1, ");
for(i=2,ROWS,V=concat(V,0);A=x*truncate(Ser(V));
for(n=0,#V-1,if(n==#V-1,V[#V]=-T(n,0));for(k=0,n, T(n,k)));print1(V[#V]", "););
print1("...]");print("");print("");print("Triangle A233530 begins:");
for(n=0,#V-2,for(k=0,n,print1(T(n,k),", "));print(""))}