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

0,5

Links

Table of n, a(n) for n=0..77.

Example

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, ...].

Prog

(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(""))}

Crossrefs

Cf. A233531, A233532, A233533, A233534, A233535 (row sums).

Sequence in context: A258170 A055830 A293109 * A079123 A121548 A180179

Adjacent sequences: A233527 A233528 A233529 * A233531 A233532 A233533

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 11 2013

Status

approved