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.
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
Offset: 1
Examples
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:
A158835 * A158831 = A158832;
A158835 * A158832 = A158833;
A158835 * A158833 = A158834;
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, ...].
Links
- Paul D. Hanna, Table of n, a(n), n = 1..1275 (rows 1..50)
- Frédéric Chapoton and Vincent Pilaud, Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra, arXiv:2201.06896 [math.CO], 2022. See p. 26.
Crossrefs
Programs
-
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 *) -
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)}
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)
Comments