A158826 -id:A158826

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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.

+10
24

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

(list; table; graph; refs; listen; history; text; internal format)

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;

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.

KEYWORD

nonn,tabl

AUTHOR

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

STATUS

approved

A121988

Number of vertices of the n-th multiplihedron.

+10
10

0, 1, 2, 6, 21, 80, 322, 1348, 5814, 25674, 115566, 528528, 2449746, 11485068, 54377288, 259663576, 1249249981, 6049846848, 29469261934, 144293491564, 709806846980, 3506278661820, 17385618278700, 86500622296800, 431718990188850, 2160826237261692

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

G.f. = x*c(x)*c(x*c(x)) where c(x) is the generating function of the Catalan numbers C(n). Thus a(n) is the Catalan transform of the sequence C(n-1). Reference for the definition of Catalan transform is the paper by Paul Barry. - Stefan Forcey (sforcey(AT)tnstate.edu), Aug 02 2007

A129442 is an essentially identical sequence. - R. J. Mathar, Jun 13 2008

From Peter Bala, Jan 27 2020: (Start)

This sequence is the main diagonal of the lower triangular array formed by putting the sequence [0, 1, 1, 2, 5, 14, 42, ...] of Catalan numbers (with 0 prepended) in the first column (k = 0) of the array and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1.

1 1

1 2 2

2 4 6 6

5 9 15 21 21

14 23 38 59 80 80

...

Cf. A307495.

Alternatively, the sequence can be obtained by multiplying the sequence of Catalan numbers by the array A106566. (End)

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..500

R. Bacher, On generating series of complementary plane trees arXiv:math/0409050 [math.CO], 2004. See p. 19.

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.

David Callan, A combinatorial interpretation of the Catalan transform of the Catalan numbers, arXiv:1111.0996 [math.CO], 2011.

Stefan Forcey, Convex Hull Realizations of the Multiplihedra, Theorem 3.2, p. 8, arXiv:0706.3226 [math.AT], 2007-2008.

Stefan Forcey, Aaron Lauve, and Frank Sottile, New Hopf Structures on Binary Trees, dmtcs:2740 - Discrete Mathematics & Theoretical Computer Science, January 1, 2009, DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009).

Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Tian-Xiao He and Louis W. Shapiro, Row sums and alternating sums of Riordan arrays, Linear Algebra and its Applications, Volume 507, 15 October 2016, Pages 77-95.

FORMULA

a(0) = 0; a(n) = C(n-1) + Sum_{i=1..n-1} a(i)*a(n-i), where C(n) = A000108(n).

G.f.: (1-sqrt(2*sqrt(1-4x)-1))/2. a(n) = (1/n)*Sum_{k=1..n} binomial(2*n-k-1,n-1)*binomial(2k-2, k-1); a(0)=0. - Stefan Forcey (sforcey(AT)tnstate.edu), Aug 02 2007

a(n) = Sum_{k = 0..n} A106566(n,k)*A000108(k-1) with A000108(-1)=0. - Philippe Deléham, Aug 27 2007

D-finite with recurrence 3*(n-1)*n*a(n) = 14*(n-1)*(2*n-3)*a(n-1) - 4*(4*n-9)*(4*n-7)*a(n-2). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ 2^(4*n-5/2)/(sqrt(Pi)*3^(n-1/2)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

G.f.: A(x) satisfies A(x)=x*(1+A(x))/((1-A(x))*(1+A(x)^3). - Vladimir Kruchinin, Jun 01 2014

G.f. is series reversion of (x - x^2) * (1 - x + x^2) = x - 2*x^2 + 2*x^3 - x^4. - Michael Somos, Jun 01 2014

From Peter Bala, Aug 22 2024: (Start)

G.f. A(x) = 1 - 1/c(x*c(x)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.

Sum_{n >= 1} a(n)*y^n = x*c(x), where y = x*(1 - x). (End)

EXAMPLE

G.f. = x + 2*x^2 + 6*x^3 + 21*x^4 + 80*x^5 + 322*x^6 + 1348*x^7 + 5814*x^8 + ...

MAPLE

a:= proc(n) option remember; `if`(n<3, n, (14*(n-1)*(2*n-3)*a(n-1)

-4*(4*n-9)*(4*n-7)*a(n-2))/ (3*n*(n-1)))

end:

seq(a(n), n=0..30); # Alois P. Heinz, Oct 20 2012

MATHEMATICA

a[0] = 0; a[n_] := a[n] = (2 n - 2)!/((n - 1)! n!) + Sum[ a[i]*a[n - i], {i, n - 1}]; Table[ a@n, {n, 0, 24}] (* Robert G. Wilson v, Jun 28 2007 *)

a[ n_] := If[ n < 1, 0, SeriesCoefficient[ InverseSeries[ Series[ x - 2 x^2 + 2 x^3 - x^4, {x, 0, n}]], {x, 0, n}]]; (* Michael Somos, Jun 01 2014 *)

a[0] = 0; a[n_] := Binomial[2n-2, n-1]*Hypergeometric2F1[1/2, 1-n, 2-2n, 4] /n; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 31 2016 *)

PROG

(PARI) {a(n) = if( n<1, 0, polcoeff( serreverse( x - 2*x^2 + 2*x^3 - x^4 + x * O(x^n)), n))}; /* Michael Somos, Jun 01 2014 */

CROSSREFS

Cf. A000108, A127632, A129442, A007317, A164965, A106566, A158826, A307495.

KEYWORD

easy,nonn

AUTHOR

Jonathan Vos Post, Jun 24 2007

EXTENSIONS

More terms from Robert G. Wilson v, Jun 28 2007

STATUS

approved

A158827

The 4th iteration of x*C(x) where C(x) is the Catalan function (A000108).

+10
4

1, 4, 20, 110, 640, 3870, 24084, 153306, 993978, 6544242, 43652340, 294469974, 2006018748, 13784115468, 95444016984, 665407010349, 4667570034444, 32922870719664, 233389493503968, 1662048903052380, 11885333877149532

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..21.

Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

FORMULA

Series reversion of 1 -4*x +12*x^2 -30*x^3 +64*x^4 -118*x^5 +188*x^6 -258*x^7 +302*x^8 -298*x^9 +244*x^10 -162*x^11 +84*x^12 -32*x^13 +8*x^14 -x^15. - R. J. Mathar, Aug 30 2021

PROG

(PARI) {a(n)=local(F=serreverse(x-x^2+O(x^(n+1))), G=x); for(i=1, 4, G=subst(F, x, G)); polcoeff(G, n)}

CROSSREFS

Cf. A158825, A158826, A158828, A158829.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 28 2009

STATUS

approved

A158828

The 5th iteration of x*C(x) where C(x) is the Catalan function (A000108).

+10
4

1, 5, 30, 195, 1330, 9380, 67844, 500619, 3755156, 28558484, 219767968, 1708590960, 13403300208, 105983648060, 844009565176, 6764300053390, 54525119251104, 441811163402124, 3597005618194848, 29412560840221272

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..20.

Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

FORMULA

Series reversion of x -5*x^2 +20*x^3 -70*x^4 +220*x^5 -630*x^6 +1656*x^7 -4014*x^8 +8994*x^9 -18654*x^10 +35832*x^11 -63750*x^12 +105024*x^13 -160120*x^14 +225696*x^15 -293685*x^16 +352074*x^17 -387820*x^18 +391232*x^19 -359992*x^20 +300664*x^21 -226580*x^22 +152952*x^23 -91656*x^24 +48204*x^25 -21924*x^26 +8456*x^27 -2692*x^28 +680*x^29 -128*x^30 +16*x^31 -x^32. - R. J. Mathar, Aug 30 2021

PROG

(PARI) {a(n)=local(F=serreverse(x-x^2+O(x^(n+1))), G=x); for(i=1, 5, G=subst(F, x, G)); polcoeff(G, n)}

CROSSREFS

Cf. A158825, A158826, A158827, A158829.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 28 2009

STATUS

approved

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).

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..24.

PROG

(PARI) {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}

CROSSREFS

Cf. A158825, A158826, A158827, A158828.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Mar 28 2009

STATUS

approved

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