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EXAMPLE
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G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 79*x^6 + 720*x^7 + 10735*x^8 + 211802*x^9 + 4968491*x^10 + 132655760*x^11 + 3943593218*x^12 +...
The table of coefficients in the successive iterations of A(x) begins:
[1, 0, 0, 0, 0, 0, 0, 0, 0, ...];
[1, 1, 2, 5, 16, 79, 720, 10735, 211802, ...];
[1, 2, 6, 21, 84, 410, 2876, 33235, 581074, ...];
[1, 3, 12, 54, 266, 1463, 9740, 90999, 1308954, ...];
[1, 4, 20, 110, 648, 4102, 28932, 248808, 2972926, ...];
[1, 5, 30, 195, 1340, 9705, 75264, 655599, 7059436, ...];
[1, 6, 42, 315, 2476, 20284, 174304, 1610487, 16952240, ...];
[1, 7, 56, 476, 4214, 38605, 366660, 3656975, 39586868, ...];
[1, 8, 72, 684, 6736, 68308, 712984, 7710392, 88021908, ...];
[1, 9, 90, 945, 10248, 114027, 1299696, 15223599, 185218134, ...];
[1, 10, 110, 1265, 14980, 181510, 2245428, 28396003, 369356822, ...]; ...
Then the triangle T=A236961 transforms the adjacent diagonals in the above table into each other, as illustrated by:
T*[1, 1, 6, 54, 648, 9705, 174304, 3656975, 88021908, ...]
= [1, 2, 12, 110, 1340, 20284, 366660, 7710392, 185218134, ...];
T*[1, 2, 12, 110, 1340, 20284, 366660, 7710392, 185218134, ...]
= [1, 3, 20, 195, 2476, 38605, 712984, 15223599, 369356822, ...];
T*[1, 3, 20, 195, 2476, 38605, 712984, 15223599, 369356822, ...]
= [1, 4, 30, 315, 4214, 68308, 1299696, 28396003, 701068918, ...]; ...
1;
1, 1;
4, 2, 1;
27, 11, 3, 1;
256, 94, 21, 4, 1;
3125, 1076, 217, 34, 5, 1;
46656, 15362, 2910, 412, 50, 6, 1;
823543, 262171, 47598, 6333, 695, 69, 7, 1;
16777216, 5198778, 915221, 116768, 12045, 1082, 91, 8, 1;
387420489, 117368024, 20182962, 2498414, 247151, 20871, 1589, 116, 9, 1;
10000000000, 2970653234, 501463686, 60678776, 5824330, 471666, 33761, 2232, 144, 10, 1; ...
such that column 0 equals A236961(n,0) = n^n.
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PROG
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(PARI) /* From Root Series G, Calculate T(n, k) of Triangle: */
{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, G +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 Root Series G and then Prints ROWS of Triangle: */
{ROWS=12; V=[1, 1]; print(""); print1("Root Sequence: [1, 1, ");
for(i=2, ROWS, V=concat(V, 0); G=x*truncate(Ser(V));
for(n=0, #V-1, if(n==#V-1, V[#V]=n^n-T(n, 0)); for(k=0, n, T(n, k))); print1(V[#V]", "); );
print1("...]"); print(""); print(""); print("Triangle begins:");
for(n=0, #V-2, for(k=0, n, print1(T(n, k), ", ")); print(""))}
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