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A158825 Square array of coefficients in the successive iterations of x*C(x) = (1-sqrt(1-4*x))/2 where C(x) is the g.f. of the Catalan numbers (A000108); read by antidiagonals. 24
1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 12, 21, 14, 1, 5, 20, 54, 80, 42, 1, 6, 30, 110, 260, 322, 132, 1, 7, 42, 195, 640, 1310, 1348, 429, 1, 8, 56, 315, 1330, 3870, 6824, 5814, 1430, 1, 9, 72, 476, 2464, 9380, 24084, 36478, 25674, 4862, 1, 10, 90, 684, 4200, 19852, 67844, 153306, 199094, 115566, 16796 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
LINKS
FORMULA
G.f. of column n = (g.f. of row n of A158830)/(1-x)^n.
Row k equals the first column of the k-th matrix power of Catalan triangle A033184; thus triangle A033184 transforms row n into row n+1 of this array (A158825). -Paul D. Hanna, Mar 30 2009
From G. C. Greubel, Apr 01 2021: (Start)
T(n, 1) = A000012(n), T(n, 2) = A000027(n).
T(n, 3) = A002378(n), T(n, 4) = A160378(n+1). (End)
EXAMPLE
Square array of coefficients in iterations of x*C(x) begins:
1, 1, 2, 5, 14, 42, 132, 429, 1430, ... A000108;
1, 2, 6, 21, 80, 322, 1348, 5814, 25674, ... A121988;
1, 3, 12, 54, 260, 1310, 6824, 36478, 199094, ... A158826;
1, 4, 20, 110, 640, 3870, 24084, 153306, 993978, ... A158827;
1, 5, 30, 195, 1330, 9380, 67844, 500619, 3755156, ... A158828;
1, 6, 42, 315, 2464, 19852, 163576, 1372196, 11682348, ...;
1, 7, 56, 476, 4200, 38052, 351792, 3305484, 31478628, ...;
1, 8, 72, 684, 6720, 67620, 693048, 7209036, 75915708, ...;
1, 9, 90, 945, 10230, 113190, 1273668, 14528217, 167607066, ...;
1, 10, 110, 1265, 14960, 180510, 2212188, 27454218, 344320262, ...;
1, 11, 132, 1650, 21164, 276562, 3666520, 49181418, 666200106, ...;
1, 12, 156, 2106, 29120, 409682, 5841836, 84218134, 1225314662, ...;
1, 13, 182, 2639, 39130, 589680, 8999172, 138755799, 2157976392, ...;
1, 14, 210, 3255, 51520, 827960, 13464752, 221101608, 3660331064, ...;
1, 15, 240, 3960, 66640, 1137640, 19640032, 342179672, 6007747368, ...;
1, 16, 272, 4760, 84864, 1533672, 28012464, 516105720, 9578580504, ...;
ILLUSTRATE ITERATIONS.
Let G(x) = x*C(x), then the first few iterations of G(x) are:
G(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + ...;
G(G(x)) = x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + ...;
G(G(G(x))) = x + 3*x^2 + 12*x^3 + 54*x^4 + 260*x^5 + ...;
G(G(G(G(x)))) = x + 4*x^2 + 20*x^3 + 110*x^4 + 640*x^5 + ...;
...
RELATED TRIANGLES.
The g.f. of column n is (g.f. of row n of A158830)/(1-x)^n
where triangle A158830 begins: 1;
1, 0;
2, 0, 0;
5, 1, 0, 0;
14, 10, 0, 0, 0;
42, 70, 8, 0, 0, 0;
132, 424, 160, 4, 0, 0, 0;
429, 2382, 1978, 250, 1, 0, 0, 0;
1430, 12804, 19508, 6276, 302, 0, 0, 0, 0;
4862, 66946, 168608, 106492, 15674, 298, 0, 0, 0, 0;
16796, 343772, 1337684, 1445208, 451948, 33148, 244, 0, 0, 0, 0;
58786, 1744314, 10003422, 16974314, 9459090, 1614906, 61806, 162, 0, 0, 0, 0;
...
Triangle A158835 transforms one diagonal into the next:
1;
1, 1;
4, 2, 1;
27, 11, 3, 1;
254, 94, 21, 4, 1;
3062, 1072, 217, 34, 5, 1;
45052, 15212, 2904, 412, 50, 6, 1;
783151, 257777, 47337, 6325, 695, 69, 7, 1; ...
so that:
where the diagonals start:
A158831 = [1, 1, 6, 54, 640, 9380, 163576, 3305484, ...];
A158832 = [1, 2, 12, 110, 1330, 19852, 351792, 7209036, ...];
A158833 = [1, 3, 20, 195, 2464, 38052, 693048, 14528217, ...];
A158834 = [1, 4, 30, 315, 4200, 67620, 1273668, 27454218, ...].
MATHEMATICA
Clear[row]; nmax = 12;
row[n_]:= row[n]= CoefficientList[Nest[(1-Sqrt[1-4#])/2&, x, n] + O[x]^(nmax+1), x] //Rest;
T[n_, k_]:= row[n][[k]];
Table[T[n-k+1, k], {n, nmax}, {k, n}]//Flatten (* Jean-François Alcover, Jul 13 2018, updated Aug 09 2018 *)
PROG
(PARI) {T(n, k)= local(F=serreverse(x-x^2+O(x^(k+2))), G=x);
for(i=1, n, G=subst(F, x, G)); polcoeff(G, k)}
CROSSREFS
Antidiagonal sums: A158829.
Related triangles: A158830, A158835.
Variant: A122888.
Sequence in context: A098474 A153199 A056860 * A247507 A107111 A082037
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 28 2009, Mar 29 2009
STATUS
approved

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Last modified March 19 06:32 EDT 2024. Contains 370953 sequences. (Running on oeis4.)