A158825 -id:A158825 - cp's OEIS frontend

A158835 Triangle, read by rows, that transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

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, 15712342, 5074738, 906557, 116372, 12035, 1082, 91, 8, 1, 357459042, 113775490, 19910808, 2483706

Offset: 1

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

Conjecture: n-th reversed row polynomial is t_n where we start with vector v of fixed length m with elements v_i = 1, then set t := v and for i=1..m-1, for j=1..i, for k=j+1..i+1 apply v_k := v_k + z*v_{k-1} and t_{i+1} := v_{i+1} (after ending each cycle for j). - Mikhail Kurkov, Sep 03 2024

			Triangle T begins:
  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;
  15712342,5074738,906557,116372,12035,1082,91,8,1;
  357459042,113775490,19910808,2483706,246596,20859,1589,116,9,1;
  9094926988,2861365660,492818850,60168736,5801510,470928,33747,2232,144,10,1;
  ...
Array A158825 of coefficients in iterations of x*C(x) begins:
  1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...;
  1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
  1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;
  1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,...;
  1,5,30,195,1330,9380,67844,500619,3755156,28558484,...;
  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,...;
  1,10,110,1265,14960,180510,2212188,27454218,344320262,...;
  ...
This triangle transforms diagonals of A158825 into each other:
T*A158831 = A158832; T*A158832 = A158833; T*A158833 = A158834;
where:
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,...].

Paul D. Hanna, Table of n, a(n), n = 1..496 (rows 1..31).

Cf. columns: A158836, A158837, A158838, A158839, row sums: A158840.

Cf. A158825, A158831, A158832, A158833, A158834, variant: A135080.

{T(n, k)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))), M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, CAT)); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))

Edited by N. J. A. Sloane, Oct 04 2010, to make entries, offset, b-file and link to b-file all consistent.

A158831 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 1, 6, 54, 640, 9380, 163576, 3305484, 75915708, 1952409954, 55573310936, 1734182983962, 58863621238500, 2159006675844616, 85088103159523296, 3585740237981536700, 160894462797493581048, 7658326127259130753070

Offset: 1

Paul D. Hanna, Mar 28 2009

nonn

Triangle A158835 transforms this sequence into A158832, the next diagonal in A158825.

			Table of coefficients in the i-th iteration of x*Catalan(x):
(1);
1,(1),2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
1,2,(6),21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,3,12,(54),260,1310,6824,36478,199094,1105478,6227712,...;
1,4,20,110,(640),3870,24084,153306,993978,6544242,43652340,...;
1,5,30,195,1330,(9380),67844,500619,3755156,28558484,...;
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),...; ...
where terms in parenthesis form the initial terms of this sequence.

Cf. A158825, A158832, A158833, A158834.

nmax = 18;
g[x_] := Module[{y}, Expand[Normal[(1 - Sqrt[1 - 4*y])/2 + O[y]^(nmax+2)] /. y -> x][[1 ;; nmax+1]]];
T = Table[Nest[g, x, n] // CoefficientList[#, x]& // Rest, {n, 1, nmax+1}];
Prepend[Diagonal[T, 1], 1] (* Jean-François Alcover, Jul 13 2018 *)

{a(n)=local(F=serreverse(x-x^2+O(x^(n+2))),G=x); for(i=1,n-1,G=subst(F,x,G));polcoeff(G,n)}

A158832 Main diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 2, 12, 110, 1330, 19852, 351792, 7209036, 167607066, 4357308098, 125219900520, 3941126688798, 134808743674176, 4979127855477336, 197480359402576304, 8370550907396970684, 377599345119560766534, 18061714498169627460982

Offset: 1

Paul D. Hanna, Mar 28 2009

nonn

Triangle A158835 transforms A158831 into this sequence, where A158831 is the previous diagonal in A158825.

Triangle A158835 transforms this sequence into A158833, the next diagonal in A158825.

			Array of coefficients in the i-th iteration of x*Catalan(x):
(1),1,2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
1,(2),6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,3,(12),54,260,1310,6824,36478,199094,1105478,6227712,...;
1,4,20,(110),640,3870,24084,153306,993978,6544242,43652340,...;
1,5,30,195,(1330),9380,67844,500619,3755156,28558484,...;
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),...;
1,10,110,1265,14960,180510,2212188,27454218,344320262,(4357308098),...; ...
where terms in parenthesis form the initial terms of this sequence.

Paul D. Hanna, Table of n, a(n), n = 1..50.

Cf. A158825, A158831, A158833, A158834.

a[n_] := Module[{x, F, G}, F = InverseSeries[x - x^2 + O[x]^(n+2)]; G = x; For[i = 1, i <= n, i++, G = (F /. x -> G)]; Coefficient[G, x, n]];
Array[a, 18] (* Jean-François Alcover, Jul 13 2018, from PARI *)

{a(n)=local(F=serreverse(x-x^2+O(x^(n+2))),G=x); for(i=1,n,G=subst(F,x,G));polcoeff(G,n)}

A158833 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 3, 20, 195, 2464, 38052, 693048, 14528217, 344320262, 9100230282, 265305808404, 8456446272144, 292528760419440, 10913859037065560, 436812586581170976, 18668379209883807385, 848499254768957476312

Offset: 1

Paul D. Hanna, Mar 28 2009

nonn

Triangle A158835 transforms A158832 into this sequence, where A158832 is the previous diagonal in A158825.

Triangle A158835 transforms this sequence into A158834, the next diagonal in A158825.

			Array of coefficients in the i-th iteration of x*Catalan(x):
1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
(1),2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,(3),12,54,260,1310,6824,36478,199094,1105478,6227712,...;
1,4,(20),110,640,3870,24084,153306,993978,6544242,43652340,...;
1,5,30,(195),1330,9380,67844,500619,3755156,28558484,...;
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,...;
1,10,110,1265,14960,180510,2212188,27454218,(344320262),...;
1,11,132,1650,21164,276562,3666520,49181418,666200106,(9100230282),...; ...
where terms in parenthesis form the initial terms of this sequence.

Cf. A158825, A158831, A158832, A158834.

a[n_] := Module[{x, F, G}, F = InverseSeries[x - x^2 + O[x]^(n+2)]; G = x; For[i = 1, i <= n+1, i++, G = (F /. x -> G)]; Coefficient[G, x, n]];
Array[a, 17] (* Jean-François Alcover, Jul 13 2018, from PARI *)

{a(n)=local(F=serreverse(x-x^2+O(x^(n+2))),G=x); for(i=1,n+1,G=subst(F,x,G));polcoeff(G,n)}

A158834 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 4, 30, 315, 4200, 67620, 1273668, 27454218, 666200106, 17968302638, 533188477536, 17261808531552, 605452449574320, 22870569475477112, 925663441858807096, 39964465820186753753, 1833332492818402014474

Offset: 1

Paul D. Hanna, Mar 28 2009

nonn

Triangle A158835 transforms A158833 into this sequence, where A158833 is the previous diagonal in A158825.

			Array of coefficients in the i-th iteration of x*Catalan(x):
1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,...;
1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
(1),3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;
1,(4),20,110,640,3870,24084,153306,993978,6544242,43652340,...;
1,5,(30),195,1330,9380,67844,500619,3755156,28558484,...;
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,...;
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,(17968302638),...; ...
where terms in parenthesis form the initial terms of this sequence.

Cf. A158825, A158831, A158832, A158833.

a[n_] := Module[{x, F, G}, F = InverseSeries[x - x^2 + O[x]^(n+2)]; G = x; For[i = 1, i <= n+2, i++, G = (F /. x -> G)]; Coefficient[G, x, n]];
Array[a, 17] (* Jean-François Alcover, Jul 13 2018, from PARI *)

{a(n)=local(F=serreverse(x-x^2+O(x^(n+2))),G=x); for(i=1,n+2,G=subst(F,x,G));polcoeff(G,n)}

A158829 Antidiagonal sums of square array A158825, in which row n lists the coefficients of the n-th iteration of x*C(x), where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 202, 861, 3972, 19648, 103500, 577443, 3396804, 20988116, 135770140, 916936351, 6449233093, 47137434787, 357331341987, 2804582808108, 22754919576652, 190578011064394, 1645490708244886, 14629351150837605

Offset: 1

Paul D. Hanna, Mar 28 2009

nonn

Cf. A158825, A158826, A158827, A158828.

{a(n)=local(F=serreverse(x-x^2+O(x^(n+1))),G=x,ADS=0); for(k=1,n,G=x;for(i=1,n-k,G=subst(F,x,G));ADS=ADS+polcoeff(G,k));ADS}

A158830 Triangle, read by rows n>=1, where row n is the n-th differences of column n of array A158825, where the g.f. of row n of A158825 is the n-th iteration of x*Catalan(x).

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Mar 28 2009

			Triangle 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;
.208012,8780912,71692452,180308420,161380816,51436848,5090124,103932,84,0,0,0,0;
....
where the g.f. of row n is (1-x)^n*[g.f. of column n of A158825];
g.f. of row n of array A158825 is the n-th iteration of x*C(x):
.1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...;
.1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
.1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;
.1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,...;
.1,5,30,195,1330,9380,67844,500619,3755156,28558484,...;
.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,...;
....
ROW-REVERSAL yields triangle A122890:
.1;
.0,1;
.0,0,2;
.0,0,1,5;
.0,0,0,10,14;
.0,0,0,8,70,42;
.0,0,0,4,160,424,132;
.0,0,0,1,250,1978,2382,429;
.0,0,0,0,302,6276,19508,12804,1430; ...
where g.f. of row n = (1-x)^n*[g.f. of column n of A122888];
g.f. of row n of A122888 is the n-th iteration of x+x^2:
.1;
.1,1;
.1,2,2,1;
.1,3,6,9,10,8,4,1;
.1,4,12,30,64,118,188,258,302,298,244,162,84,32,8,1; ...

Paul D. Hanna, Table of n, a(n), n = 1..1326 (rows 1..51).
Toufik Mansour, Howard Skogman, Rebecca Smith, Passing through a stack k times, arXiv:1704.04288 [math.CO], 2017.

Cf. A158825, A122890 (row-reversal), A122888, columns: A000108, A122892.

nmax = 11;
f[0][x_] := x; f[n_][x_] := f[n][x] = f[n - 1][x + x^2] // Expand;
T = Table[SeriesCoefficient[f[n][x], {x, 0, k}], {n, 0, nmax}, {k, 1, nmax}];
row[n_] := CoefficientList[(1-x)^n*(T[[All, n]].x^Range[0, nmax])+O[x]^nmax, x] // Reverse;
Table[row[n], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Oct 26 2018 *)

{T(n, k)=local(F=x, CAT=serreverse(x-x^2+x*O(x^(n+2))), M, N, P); M=matrix(n+2, n+2, r, c, F=x; for(i=1, r, F=subst(F, x, CAT)); polcoeff(F, c)); Vec(truncate(Ser(vector(n+1,r,M[r,n+1])))*(1-x)^(n+1) +x*O(x^k))[k+1]}

Row sums equal the factorial numbers.
G.f. of row n = (1-x)^n*[g.f. of column n of A158825] where the g.f. of row n of array A158825 is the n-th iteration of x*C(x) and C(x) is the g.f. of the Catalan sequence A000108.
Row-reversal is triangle A122890 where g.f. of row n of A122890 = (1-x)^n*[g.f. of column n of A122888], and the g.f. of row n of array A122888 is the n-th iteration of x+x^2.

Edited by N. J. A. Sloane, Oct 04 2010, to make entries, offset, b-file and link to b-file all consistent.

A166905 Triangle, read by rows, that transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).

Original entry on oeis.org

1, 1, 1, 6, 4, 1, 54, 33, 9, 1, 640, 380, 108, 16, 1, 9380, 5510, 1610, 270, 25, 1, 163576, 95732, 28560, 5148, 570, 36, 1, 3305484, 1933288, 586320, 110929, 13650, 1071, 49, 1, 75915708, 44437080, 13658904, 2677008, 353600, 31624, 1848, 64, 1, 1952409954

Offset: 0

Paul D. Hanna, Nov 28 2009

			Triangle begins:
1;
1,1;
6,4,1;
54,33,9,1;
640,380,108,16,1;
9380,5510,1610,270,25,1;
163576,95732,28560,5148,570,36,1;
3305484,1933288,586320,110929,13650,1071,49,1;
75915708,44437080,13658904,2677008,353600,31624,1848,64,1;
1952409954,1144564278,355787568,71648322,9962949,973845,66150,2988,81,1;
55573310936,32638644236,10243342296,2107966432,304857190,31795560,2395120,127720,4590,100,1;
...
Coefficients in iterations of x*Catalan(x) form table A158825:
1,1,2,5,14,42,132,429,1430,4862,16796,58786,208012,742900,...;
1,2,6,21,80,322,1348,5814,25674,115566,528528,2449746,...;
1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,...;
1,4,20,110,640,3870,24084,153306,993978,6544242,43652340,...;
1,5,30,195,1330,9380,67844,500619,3755156,28558484,...;
1,6,42,315,2464,19852,163576,1372196,11682348,100707972,...;
1,7,56,476,4200,38052,351792,3305484,31478628,303208212,...;
...
This triangle T transforms rows into diagonals of A158825;
the initial diagonals begin:
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,...].
For example:
T * [1,0,0,0,0,0,0,0,0,0,0,0,0, ...] = A158831;
T * [1,1,2,5,14,42,132,429,1430,...] = A158832;
T * [1,2,6,21,80,322,1348,5814, ...] = A158833;
T * [1,3,12,54,260, 1310, 6824, ...] = A158834.

Cf. A166906, A166907, A166908, A166909, variant: A166900.

Cf. A158825, A158831, A158832, A158833, A158834, A158835, A000108.

{T(n, k)=local(F=x, G=serreverse(x-x^2+O(x^(n+3))), M, N, P, m=n); M=matrix(m+2, m+2, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, G+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, F=x;for(i=1, r, F=subst(F, x, G+x*O(x^(m+2)))); polcoeff(F, c)); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}

A122890 Triangle, read by rows, where the g.f. of row n divided by (1-x)^n yields the g.f. of column n in the triangle A122888, for n>=1.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 5, 0, 0, 0, 10, 14, 0, 0, 0, 8, 70, 42, 0, 0, 0, 4, 160, 424, 132, 0, 0, 0, 1, 250, 1978, 2382, 429, 0, 0, 0, 0, 302, 6276, 19508, 12804, 1430, 0, 0, 0, 0, 298, 15674, 106492, 168608, 66946, 4862, 0, 0, 0, 0, 244, 33148, 451948, 1445208, 1337684, 343772, 16796

Offset: 0

Paul D. Hanna, Sep 18 2006

Main diagonal forms the Catalan numbers (A000108). Row sums gives the factorials. In table A122888, row n lists the coefficients of x^k, k = 1..2^n, in the n-th self-composition of (x + x^2) for n >= 0.

Parker gave the following combinatorial interpretation of the numbers: For n > 0, T(n, j) is the number of sequences c_1c_2...c_n of positive integers such that 1 <= c_i <= i for each i in {1, 2, .., n} with exactly j - 1 values of i such that c_i <= c_{i+1}. - Peter Luschny, May 05 2013

			Triangle begins:
1;
0,1;
0,0,2;
0,0,1,5;
0,0,0,10,14;
0,0,0,8,70,42;
0,0,0,4,160,424,132;
0,0,0,1,250,1978,2382,429;
0,0,0,0,302,6276,19508,12804,1430;
0,0,0,0,298,15674,106492,168608,66946,4862;
0,0,0,0,244,33148,451948,1445208,1337684,343772,16796;
0,0,0,0,162,61806,1614906,9459090,16974314,10003422,1744314,58786;
0,0,0,0,84,103932,5090124,51436848,161380816,180308420,71692452,8780912,208012; ...
Table A122888 starts:
1;
1, 1;
1, 2, 2, 1;
1, 3, 6, 9, 10, 8, 4, 1;
1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...;
where row n gives the g.f. of the n-th self-composition of (x+x^2).
From _Paul D. Hanna_, Apr 11 2009: (Start)
ROW-REVERSAL yields triangle A158830:
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; ...
where
g.f. of row n of A158830 = (1-x)^n*[g.f. of column n of A158825];
g.f. of row n of A158825 = n-th iteration of x*Catalan(x).
RELATED ARRAY A158825 begins:
1,1,2,5,14,42,132,429,1430,4862,16796,58786,...;
1,2,6,21,80,322,1348,5814,25674,115566,528528,...;
1,3,12,54,260,1310,6824,36478,199094,1105478,...;
1,4,20,110,640,3870,24084,153306,993978,...;
1,5,30,195,1330,9380,67844,500619,3755156,...;
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,...; ...
which consists of successive iterations of x*Catalan(x).
(End)

Toufik Mansour, Mark Shattuck, Statistics on bargraphs of inversion sequences of permutations, Discrete Math. Lett. (2020) Vol. 4, 42-49.
Toufik Mansour, Howard Skogman, Rebecca Smith, Passing through a stack k times, arXiv:1704.04288 [math.CO], 2017.
Susan Field Parker, The Combinatorics of Functional Composition and Inversion, Ph.D. Dissertation, Brandeis Univ. (1993) (Section 2.3.4, p. 27,28.)

Cf. A122888; A122891 (column sums); diagonals: A122892, A000108.

Cf. related tables: A158830, A158825. [Paul D. Hanna, Apr 11 2009]

nmax = 11;
f[0][x_] := x; f[n_][x_] := f[n][x] = f[n - 1][x + x^2] // Expand; T = Table[ SeriesCoefficient[f[n][x], {x, 0, k}], {n, 0, nmax}, {k, 1, nmax}];
row[n_] := CoefficientList[(1-x)^n*(T[[All, n]].x^Range[0, nmax])+O[x]^nmax, x];
Table[row[n], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Jul 13 2018 *)

From Paul D. Hanna, Apr 11 2009: (Start)
G.f. of row n: (1-x)^n*[g.f. of column n of A122888] where the g.f. of row n of A122888 is the n-th iteration of x+x^2.
Row-reversal forms triangle A158830 where g.f. of row n of A158830 = (1-x)^n*[g.f. of column n of A158825], and the g.f. of row n of array A158825 is the n-th iteration of x*C(x) and C(x) is the g.f. of the Catalan sequence A000108. (End)

A158826 Third iteration of x*C(x) where C(x) is the Catalan function (A000108).

Original entry on oeis.org

1, 3, 12, 54, 260, 1310, 6824, 36478, 199094, 1105478, 6227712, 35520498, 204773400, 1191572004, 6990859416, 41313818217, 245735825082, 1470125583756, 8840948601024, 53417237877396, 324123222435804, 1974317194619712

Offset: 1

Paul D. Hanna, Mar 28 2009

nonn

Series reversion of x - 3*x^2 + 6*x^3 - 9*x^4 + 10*x^5 - 8*x^6 + 4*x^7 - x^8. - Benedict W. J. Irwin, Oct 19 2016

Column 1 of A106566^3 (see Barry, Section 3). - Peter Bala, Apr 11 2017

G. C. Greubel, Table of n, a(n) for n = 1..1000
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5, pp. 1-24.
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A121988 (2nd), A158825, A158827 (4th), A158828, A158829.

max = 22; c[x_] := Sum[ CatalanNumber[n]*x^n, {n, 0, max}]; f[x_] := x*c[x]; CoefficientList[ Series[ f@f@f@x, {x, 0, max}], x] // Rest (* Jean-François Alcover, Jan 24 2013 *)
Rest@CoefficientList[InverseSeries[x-3x^2+6x^3-9x^4+10x^5-8x^6+4x^7-x^8+O[x]^30], x] (* Benedict W. J. Irwin, Oct 19 2016 *)

a(n):=sum(binomial(2*k-2,k-1)*sum(binomial(-k+2*i-1,i-1)*binomial(2*n-i-1,n-1),i,k,n),k,1,n)/n; /* Vladimir Kruchinin, Jan 24 2013 */

a(n)=local(F=serreverse(x-x^2+O(x^(n+1))),G=x); for(i=1,3,G=subst(F,x,G)); polcoeff(G,n)

from sympy import binomial as C
def a(n):
    return sum(C(2*k - 2, k - 1) * sum(C(-k + 2*i - 1, i - 1) * C(2*n - i - 1, n - 1) for i in range(k, n + 1)) for k in range(1, n + 1)) / n
[a(n) for n in range(1, 51)] # Indranil Ghosh, Apr 12 2017

a(n) = (1/n)*Sum_{k=1..n} [ binomial(2*k-2,k-1)*Sum_{i=k..n}( binomial(-k+2*i-1,i-1)*binomial(2*n-i-1,n-1) ) ]. - Vladimir Kruchinin, Jan 24 2013
G.f.: (1 - sqrt(-1 + 2*sqrt(-1 + 2*sqrt(1 - 4*x))))/2. - Benedict W. J. Irwin, Oct 19 2016
a(n) ~ 2^(8*n - 3) / (sqrt(5*Pi) * n^(3/2) * 39^(n - 1/2)). - Vaclav Kotesovec, Jul 20 2019
Conjecture D-finite with recurrence 1053*n*(n-1)*(n-2)*(n-3)*a(n) -36*(n-1)*(n-2)*(n-3)*(634*n-1367)*a(n-1) +24*(n-2)*(n-3)*(7966*n^2-43500*n+61181)*a(n-2) -8*(n-3)*(96128*n^3-957424*n^2+3221878*n-3665189)*a(n-3) +16*(91904*n^4-1446528*n^3+8575792*n^2-22703688*n+22652013)*a(n-4) -256*(8*n-35)*(8*n-41)*(8*n-39)*(8*n-37)*a(n-5)=0. - R. J. Mathar, Aug 30 2021

cp's OEIS Frontend

A158835 Triangle, read by rows, that transforms diagonals in the array A158825 of coefficients of successive iterations of x*C(x) where C(x) is the Catalan function (A000108).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A158831 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A158832 Main diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A158833 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A158834 A diagonal in the array A158825 of coefficients of successive iterations of x*C(x), where C(x) is the Catalan function (A000108).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A158829 Antidiagonal sums of square array A158825, in which row n lists the coefficients of the n-th iteration of x*C(x), where C(x) is the Catalan function (A000108).

Views

Author

Keywords

Crossrefs

Programs

PARI

A158830 Triangle, read by rows n>=1, where row n is the n-th differences of column n of array A158825, where the g.f. of row n of A158825 is the n-th iteration of x*Catalan(x).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A166905 Triangle, read by rows, that transforms rows into diagonals in the table A158825 of coefficients in successive iterations of x*Catalan(x) (cf. A000108).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI