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EXAMPLE
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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, ...].
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PROG
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(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(""))}
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