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

1,5

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.

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:
  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, ...].

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

Rows: A000108, A121988, A158826, A158827, A158828.

Columns: A000012, A000027, A002378, A160378.

Antidiagonal sums: A158829.

Diagonals: A158831, A158832, A158833, A158834.

Related triangles: A158830, A158835.

Variant: A122888.

Sequence in context: A098474 A153199 A056860 * A247507 A107111 A082037

Adjacent sequences: A158822 A158823 A158824 * A158826 A158827 A158828

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 28 2009, Mar 29 2009

Status

approved