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%I A158825 #16 May 18 2024 14:53:51
%S A158825 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,
%T A158825 132,1,7,42,195,640,1310,1348,429,1,8,56,315,1330,3870,6824,5814,1430,
%U A158825 1,9,72,476,2464,9380,24084,36478,25674,4862,1,10,90,684,4200,19852,67844,153306,199094,115566,16796
%N 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.
%H A158825 Paul D. Hanna, Table of n, a(n), n = 1..1275 (rows 1..50)
%H A158825 Frédéric Chapoton and Vincent Pilaud, Shuffles of deformed permutahedra, multiplihedra, constrainahedra, and biassociahedra, arXiv:2201.06896 [math.CO], 2022. See p. 26.
%F A158825 G.f. of column n = (g.f. of row n of A158830)/(1-x)^n.
%F A158825 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
%F A158825 From _G. C. Greubel_, Apr 01 2021: (Start)
%F A158825 T(n, 1) = A000012(n), T(n, 2) = A000027(n).
%F A158825 T(n, 3) = A002378(n), T(n, 4) = A160378(n+1). (End)
%e A158825 Square array of coefficients in iterations of x*C(x) begins:
%e A158825 1, 1, 2, 5, 14, 42, 132, 429, 1430, ... A000108;
%e A158825 1, 2, 6, 21, 80, 322, 1348, 5814, 25674, ... A121988;
%e A158825 1, 3, 12, 54, 260, 1310, 6824, 36478, 199094, ... A158826;
%e A158825 1, 4, 20, 110, 640, 3870, 24084, 153306, 993978, ... A158827;
%e A158825 1, 5, 30, 195, 1330, 9380, 67844, 500619, 3755156, ... A158828;
%e A158825 1, 6, 42, 315, 2464, 19852, 163576, 1372196, 11682348, ...;
%e A158825 1, 7, 56, 476, 4200, 38052, 351792, 3305484, 31478628, ...;
%e A158825 1, 8, 72, 684, 6720, 67620, 693048, 7209036, 75915708, ...;
%e A158825 1, 9, 90, 945, 10230, 113190, 1273668, 14528217, 167607066, ...;
%e A158825 1, 10, 110, 1265, 14960, 180510, 2212188, 27454218, 344320262, ...;
%e A158825 1, 11, 132, 1650, 21164, 276562, 3666520, 49181418, 666200106, ...;
%e A158825 1, 12, 156, 2106, 29120, 409682, 5841836, 84218134, 1225314662, ...;
%e A158825 1, 13, 182, 2639, 39130, 589680, 8999172, 138755799, 2157976392, ...;
%e A158825 1, 14, 210, 3255, 51520, 827960, 13464752, 221101608, 3660331064, ...;
%e A158825 1, 15, 240, 3960, 66640, 1137640, 19640032, 342179672, 6007747368, ...;
%e A158825 1, 16, 272, 4760, 84864, 1533672, 28012464, 516105720, 9578580504, ...;
%e A158825 ILLUSTRATE ITERATIONS.
%e A158825 Let G(x) = x*C(x), then the first few iterations of G(x) are:
%e A158825 G(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + ...;
%e A158825 G(G(x)) = x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + ...;
%e A158825 G(G(G(x))) = x + 3*x^2 + 12*x^3 + 54*x^4 + 260*x^5 + ...;
%e A158825 G(G(G(G(x)))) = x + 4*x^2 + 20*x^3 + 110*x^4 + 640*x^5 + ...;
%e A158825 ...
%e A158825 RELATED TRIANGLES.
%e A158825 The g.f. of column n is (g.f. of row n of A158830)/(1-x)^n
%e A158825 where triangle A158830 begins: 1;
%e A158825 1, 0;
%e A158825 2, 0, 0;
%e A158825 5, 1, 0, 0;
%e A158825 14, 10, 0, 0, 0;
%e A158825 42, 70, 8, 0, 0, 0;
%e A158825 132, 424, 160, 4, 0, 0, 0;
%e A158825 429, 2382, 1978, 250, 1, 0, 0, 0;
%e A158825 1430, 12804, 19508, 6276, 302, 0, 0, 0, 0;
%e A158825 4862, 66946, 168608, 106492, 15674, 298, 0, 0, 0, 0;
%e A158825 16796, 343772, 1337684, 1445208, 451948, 33148, 244, 0, 0, 0, 0;
%e A158825 58786, 1744314, 10003422, 16974314, 9459090, 1614906, 61806, 162, 0, 0, 0, 0;
%e A158825 ...
%e A158825 Triangle A158835 transforms one diagonal into the next:
%e A158825 1;
%e A158825 1, 1;
%e A158825 4, 2, 1;
%e A158825 27, 11, 3, 1;
%e A158825 254, 94, 21, 4, 1;
%e A158825 3062, 1072, 217, 34, 5, 1;
%e A158825 45052, 15212, 2904, 412, 50, 6, 1;
%e A158825 783151, 257777, 47337, 6325, 695, 69, 7, 1; ...
%e A158825 so that:
%e A158825 A158835 * A158831 = A158832;
%e A158825 A158835 * A158832 = A158833;
%e A158825 A158835 * A158833 = A158834;
%e A158825 where the diagonals start:
%e A158825 A158831 = [1, 1, 6, 54, 640, 9380, 163576, 3305484, ...];
%e A158825 A158832 = [1, 2, 12, 110, 1330, 19852, 351792, 7209036, ...];
%e A158825 A158833 = [1, 3, 20, 195, 2464, 38052, 693048, 14528217, ...];
%e A158825 A158834 = [1, 4, 30, 315, 4200, 67620, 1273668, 27454218, ...].
%t A158825 Clear[row]; nmax = 12;
%t A158825 row[n_]:= row[n]= CoefficientList[Nest[(1-Sqrt[1-4#])/2&, x, n] + O[x]^(nmax+1), x] //Rest;
%t A158825 T[n_, k_]:= row[n][[k]];
%t A158825 Table[T[n-k+1, k], {n, nmax}, {k, n}]//Flatten (* _Jean-François Alcover_, Jul 13 2018, updated Aug 09 2018 *)
%o A158825 (PARI) {T(n,k)= local(F=serreverse(x-x^2+O(x^(k+2))), G=x);
%o A158825 for(i=1, n, G=subst(F,x,G)); polcoeff(G,k)}
%Y A158825 Rows: A000108, A121988, A158826, A158827, A158828.
%Y A158825 Columns: A000012, A000027, A002378, A160378.
%Y A158825 Antidiagonal sums: A158829.
%Y A158825 Diagonals: A158831, A158832, A158833, A158834.
%Y A158825 Related triangles: A158830, A158835.
%Y A158825 Variant: A122888.
%K A158825 nonn,tabl
%O A158825 1,5
%A A158825 _Paul D. Hanna_, Mar 28 2009, Mar 29 2009
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