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Paul D. Hanna, <a href="/A158825/b158825.txt">Table of n, a(n), n = 1..1275 (rows 1..50).</a>

Frédéric Chapoton and Vincent Pilaud, <a href="https://arxiv.org/abs/2201.06896">Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra</a>, arXiv:2201.06896 [math.CO], 2022. See p. 26.

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

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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

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). [From _-_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)

1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...;

1, 1, 2, 5, 14, 42, 132, 429, 1430, ... A000108;

1, 2, 6, 21, 80, 322, 1348, 5814, 25674,115566,528528,2449746,11485068, ... A121988;

1, 3, 12, 54, 260, 1310, 6824, 36478, 199094,1105478,6227712,35520498, ... A158826;

1, 4, 20, 110, 640, 3870, 24084, 153306, 993978,6544242,43652340, ... A158827;

1, 5, 30, 195, 1330, 9380, 67844, 500619, 3755156,28558484,219767968, ... A158828;

1, 6, 42, 315, 2464, 19852, 163576, 1372196, 11682348,100707972, ...;

1, 7, 56, 476, 4200, 38052, 351792, 3305484, 31478628,303208212, ...;

1, 8, 72, 684, 6720, 67620, 693048, 7209036, 75915708,807845676, ...;

1, 9, 90, 945, 10230, 113190, 1273668, 14528217, 167607066,1952409954, ...;

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, ...; ...

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 + 42*x^6 + 132*x^7 +...;

G(G(x)) = x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + 322*x^6 +...;

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 + ...;

The g.f. of column n is [(g.f. of row n of A158830])/(1-x)^n

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;

...

...

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; ...

A158835 * A158831 = A158832;

A158835 * A158832 = A158833;

A158835 * A158833 = A158834;

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, ...].

Clear[row]; nmax = 12;

Clear[row]; 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, 1, nmax}, {k, 1, 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)}

for(i=1, n, G=subst(F, x, G)); polcoeff(G, k)}

Cf. rowsRows: A000108, A121988, A158826, A158827, A158828; antidiagonal sums: A158829.

Cf. diagonalsColumns: A158831, A158832, A158833, A158834A000012, A000027, A002378, A160378.

Antidiagonal sums: A158829.

Diagonals: A158831, A158832, A158833, A158834.

Cf. related Related triangles: A158830, A158835, variant: A122888.

Variant: A122888.

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gClear[row]; row[x_n_] := Modulerow[{y}, Expandn] = CoefficientList[NormalNest[(1-Sqrt[1-4*y#])/2 &, x, n] + O[y]^(nmax+2)] /. y -> x][[1 ; ; ^(nmax+1]] ), x] // Rest;

T[n_, k_] := row[n][[k]];

T = Table[NestT[g, x, n] // CoefficientList[#, x-k+1, k]& // Rest, , {n, 1, nmax+}, {k, 1, n}]; // Flatten (* _Jean-François Alcover_, Jul 13 2018, updated Aug 09 2018 *)

Table[T[[n-k+1, k]], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 13 2018 *)

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