A233530 Triangle, read by rows, that transforms diagonals in the table of coefficients in the successive iterations of the g.f. (A233531) such that column 0 consists of all zeros 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

Offset: 0

Author

Paul D. Hanna, Dec 11 2013

Keywords

• nonn
• tabl

Examples

			Triangle begins:
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;
0, 4008441848, 2943137604, 974898636, 202185010, 30319020, 3572037, 349720, 29718, 2280, 165, 12, 1; ...
in which column 0 consists of all zeros after row 1.
ILLUSTRATION OF GENERATING METHOD.
The g.f. of A233531 begins:
G(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 - 104538530634844*x^17 - 4010605941377292*x^18 +...
If we form a table of coefficients in the iterations of G(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 this triangle T 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. A233531, A233532, A233533, A233534, A233535 (row sums).

Programs

PARI

/* Given 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]=-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(""))}