A129442 -id:A129442 - cp's OEIS frontend

A121988 Number of vertices of the n-th multiplihedron.

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

Offset: 0

Jonathan Vos Post, Jun 24 2007

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.
    0
    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)

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

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.

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

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

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 *)

{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 */

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

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

A381831 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) * (1-x+x^2))^3 ) )^(1/3).

Original entry on oeis.org

1, 2, 14, 133, 1456, 17306, 217066, 2827896, 37895130, 519000037, 7232429952, 102220846756, 1461817707558, 21112968248198, 307527937374182, 4512344039147420, 66634574697351360, 989569163283434676, 14769533757869187052, 221426909287107012800, 3333042591222552282784, 50353576994047154278451

Offset: 0

Seiichi Manyama, Mar 08 2025

nonn

Cf. A364592, A381818, A381830.
Cf. A129442, A381828.
Cf. A000108.

my(N=30, x='x+O('x^N)); Vec((serreverse(x*((1-x)*(1-x+x^2))^3)/x)^(1/3))

G.f. A(x) satisfies A(x) = C(x*A(x)^2) / (1 - x*A(x)^3), where C(x) is the g.f. of A000108.

a(n) = Sum_{k=0..n} binomial(3*n+k+1,k) * binomial(4*n-2*k,n-k)/(3*n+k+1).

A381877 G.f. A(x) satisfies A(x) = C(x) / (1 - x*A(x))^4, where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 5, 36, 307, 2891, 29029, 304716, 3303712, 36708842, 415818822, 4783832314, 55743318579, 656528284027, 7802975428711, 93467830304056, 1127239608233884, 13676060532043690, 166800618473750824, 2043978275887704674, 25152767272402722288, 310703538187552229521

Offset: 0

Seiichi Manyama, Mar 09 2025

nonn

Cf. A129442, A381875, A381876.

Cf. A000108, A381861.

a(n) = sum(k=0, n, binomial(n+k+1, k)*binomial(5*n-5*k+3, n-k)/(n+k+1));

a(n) = Sum_{k=0..n} binomial(n+k+1,k) * binomial(5*n-5*k+3,n-k)/(n+k+1).

a(n) = binomial(3 + 5*n, n)*hypergeom([-3/4-n, -1/2-n, -1/4-n, -n, 1+n], [-3/5-n, -2/5-n, -1/5-n, 1/5-n], 2^8/5^5)/(1 + n). - Stefano Spezia, Mar 09 2025

A127631 Square of Riordan array (1, x*c(x)) where c(x) is the g.f. of A000108.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 21, 16, 6, 1, 0, 80, 66, 30, 8, 1, 0, 322, 280, 143, 48, 10, 1, 0, 1348, 1216, 672, 260, 70, 12, 1, 0, 5814, 5385, 3150, 1344, 425, 96, 14, 1, 0, 25674, 24244, 14799, 6784, 2400, 646, 126, 16, 1

Offset: 0

Paul Barry, Jan 20 2007

Square of A106566. Row sums are A127632.

			Triangle begins
  1;
  0,      1;
  0,      2,      1;
  0,      6,      4,     1;
  0,     21,     16,     6,     1;
  0,     80,     66,    30,     8,     1;
  0,    322,    280,   143,    48,    10,    1;
  0,   1348,   1216,   672,   260,    70,   12,   1;
  0,   5814,   5385,  3150,  1344,   425,   96,  14,   1;
  0,  25674,  24244, 14799,  6784,  2400,  646, 126,  16,  1;
  0, 115566, 110704, 69828, 33814, 13002, 3960, 931, 160, 18, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A106566, A121988, A129442.

[[k eq n select 1 else (k/n)*(&+[Binomial(2*j+k-1,j)*Binomial(2*n -k-j-1, n-k-j): j in [0..n-k]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 05 2019

T[n_, k_]:= If[k==n, 1, (k/n)*Sum[Binomial[2*j-k-1, j-k]*Binomial[2*n-j- 1, n-j], {j,k,n}]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 05 2019 *)

T(n,k):=if k=n then 1 else if n=0 then 0 else (k*sum((binomial(-k+2*i-1,i-k))*(binomial(2*n-i-1,n-i)),i,k,n))/n; /* Vladimir Kruchinin, Apr 05 2019 */

{T(n,k) = if(k==n, 1, (k/n)*sum(j=0,n-k, binomial(2*j+k-1, j)* binomial(2*n-k-j-1, n-k-j)))}; \\ G. C. Greubel, Apr 05 2019

def T(n, k):
   if k == n: return 1
   return (k*sum(binomial(2*j+k-1, j)* binomial(2*n-k-j-1, n-k-j) for j in (0..n-k)))//n
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 05 2019

Riordan array (1, x*c(x)*c(x*c(x))), where c(x) is the g.f. of A000108.
T(n+1,1) = A129442(n) = A121988(n+1). - Philippe Deléham, Feb 27 2013
T(n,k) = (k/n)*Sum_{i=k..n} C(2*i-k-1,i-k)*C(2*n-i-1,n-i), T(n,n)=1. - Vladimir Kruchinin, Apr 05 2019

A368935 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x+x^3) ).

Original entry on oeis.org

1, 2, 7, 29, 132, 637, 3200, 16554, 87576, 471570, 2575885, 14238003, 79487023, 447540164, 2538352756, 14489355578, 83174465721, 479842193453, 2780625587824, 16178040713569, 94467163314370, 553430174678595, 3251969073086610, 19161172609833540, 113186247571818096

Offset: 0

Seiichi Manyama, Jan 10 2024

nonn

Cf. A129442, A368936, A368937.

Cf. A063033, A368931.

a(n) = sum(k=0, n\3, (-1)^k*binomial(n+k, k)*binomial(3*n-2*k+1, n-3*k))/(n+1);

my(x='x+O('x^30)); Vec(serreverse(x*(1-x)*(1-x+x^3))/x) \\ Michel Marcus, Jan 10 2024

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (-1)^k * binomial(n+k,k) * binomial(3*n-2*k+1,n-3*k).

A368936 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x+x^4) ).

Original entry on oeis.org

1, 2, 7, 30, 142, 715, 3756, 20349, 112865, 637681, 3657075, 21233199, 124562708, 737197980, 4396176336, 26389742175, 159336837840, 967007923321, 5895699043010, 36093405644877, 221785663880176, 1367420967329725, 8456765007380190, 52447676008911675

Offset: 0

Seiichi Manyama, Jan 10 2024

nonn

Cf. A129442, A368935, A368937.

Cf. A368932.

a(n) = sum(k=0, n\4, (-1)^k*binomial(n+k, k)*binomial(3*n-3*k+1, n-4*k))/(n+1);

my(x='x+O('x^30)); Vec(serreverse(x*(1-x)*(1-x+x^4))/x) \\ Michel Marcus, Jan 10 2024

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (-1)^k * binomial(n+k,k) * binomial(3*n-3*k+1,n-4*k).

A368937 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x+x^5) ).

Original entry on oeis.org

1, 2, 7, 30, 143, 727, 3861, 21165, 118845, 680064, 3951291, 23247874, 138229486, 829292780, 5013767772, 30516496017, 186837457296, 1149894814718, 7110026033305, 44146396259805, 275139524189497, 1720647439298395, 10793938343564655, 67905034046934225

Offset: 0

Seiichi Manyama, Jan 10 2024

nonn

Cf. A129442, A368935, A368936.

Cf. A366046, A368933.

a(n) = sum(k=0, n\5, (-1)^k*binomial(n+k, k)*binomial(3*n-4*k+1, n-5*k))/(n+1);

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)*(1-x+x^5))/x)

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/5)} (-1)^k * binomial(n+k,k) * binomial(3*n-4*k+1,n-5*k).

A381828 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) * (1-x+x^2))^2 ) )^(1/2).

Original entry on oeis.org

1, 2, 10, 65, 480, 3824, 32039, 278256, 2482578, 22617830, 209540672, 1968031520, 18696064179, 179332892186, 1734451272240, 16895744042472, 165621305486976, 1632518433458400, 16170959983623314, 160888256475481560, 1607061512154585046, 16110030923830784248

Offset: 0

Seiichi Manyama, Mar 08 2025

nonn

Cf. A188687, A381817, A381829.
Cf. A129442, A381831.
Cf. A000108, A368975.

my(N=30, x='x+O('x^N)); Vec((serreverse(x*((1-x)*(1-x+x^2))^2)/x)^(1/2))

G.f. A(x) satisfies A(x) = C(x*A(x)) / (1 - x*A(x)^2), where C(x) is the g.f. of A000108.
a(n) = Sum_{k=0..n} binomial(2*n+k+1,k) * binomial(3*n-2*k,n-k)/(2*n+k+1).
D-finite with recurrence +432*n*(n-1)*(n-2)*(2*n+1)*(2*n-1)*(2*n-3)*(262261060139434887136491*n -880264534325728808928710)*a(n) +24*(n-1)*(n-2)*(2*n-1)*(2*n-3)*(9441398165019655936913676*n^3 -1563359509176097527827297363*n^2 +8122005300033248841454135898*n -10005843136737488906545668303)*a(n-1) -8*(n-2)*(2*n-3)*(26904862014415612504704360259*n^5 -439294650192331167438487778367*n^4 +2462557164881954865201862193560*n^3 -6116391863054255517662202621591*n^2 +6730597164009721987374566778403*n -2508886036978141982914230533400)*a(n-2) +2*(3280856375160701992555505608813*n^7 -60505233834440544774094319915261*n^6 +458650706405377012453301766859297*n^5 -1843996542698657351167896639498197*n^4 +4199211312282774397146042070543498*n^3 -5283107978583820687249123910721062*n^2 +3195330463869279708956264243293272*n -571272270914692694572799416918200)*a(n-3) +3*(-10499174187769013704183946812135*n^7 +189831332911960443054698384732480*n^6 -1395267797131742288585801071743534*n^5 +5221938509132769354051685228032464*n^4 -9839826026184653630837080778918103*n^3 +6229383740555425356174546560814416*n^2 +6216439623275682391743799709941612*n -8390747283534155728971424365124320)*a(n-4) -112*(7*n-31)*(7*n-32) *(2094251874056865218841652*n -5622141652266976856940223)*(7*n-29)*(7*n-26) *(7*n-30)*(7*n-27)*a(n-5)=0. - R. J. Mathar, Mar 10 2025

A381875 G.f. A(x) satisfies A(x) = C(x) / (1 - x*A(x))^2, where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 3, 13, 66, 368, 2185, 13570, 87147, 574241, 3861286, 26390591, 182798850, 1280387583, 9053335674, 64534088960, 463249047099, 3345832486407, 24296575830677, 177286818019264, 1299208549351640, 9557974679439901, 70563100013789595, 522608148884843970

Offset: 0

Seiichi Manyama, Mar 09 2025

nonn

Cf. A129442, A381876, A381877.

Cf. A000108.

a(n) = sum(k=0, n, binomial(n+k+1, k)*binomial(3*n-3*k+1, n-k)/(n+k+1));

a(n) = Sum_{k=0..n} binomial(n+k+1,k) * binomial(3*n-3*k+1,n-k)/(n+k+1).

a(n) = binomial(1 + 3*n, n)*hypergeom([-1/2-n, -n, 1+n], [-1/3-n, 1/3-n], 2^2/3^3)/(1 + n). - Stefano Spezia, Mar 09 2025

A381876 G.f. A(x) satisfies A(x) = C(x) / (1 - x*A(x))^3, where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 4, 23, 156, 1167, 9311, 77710, 670294, 5928183, 53467931, 489904745, 4547296624, 42667426369, 404044679434, 3856480309376, 37062228265769, 358330619946164, 3482936427997599, 34014454418349579, 333598711996924548, 3284326412065118717, 32446900771699499147

Offset: 0

Seiichi Manyama, Mar 09 2025

nonn

Cf. A129442, A381875, A381877.

Cf. A000108, A381880.

a(n) = sum(k=0, n, binomial(n+k+1, k)*binomial(4*n-4*k+2, n-k)/(n+k+1));

a(n) = Sum_{k=0..n} binomial(n+k+1,k) * binomial(4*n-4*k+2,n-k)/(n+k+1).

a(n) = binomial(2 + 4*n, n)*hypergeom([-2/3-n, -1/3-n, -n, 1+n], [-1/2-n, -1/4-n, 1/4-n], 3^3/2^8)/(1 + n). - Stefano Spezia, Mar 09 2025

cp's OEIS Frontend

A121988 Number of vertices of the n-th multiplihedron.

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A381831 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) * (1-x+x^2))^3 ) )^(1/3).

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A381877 G.f. A(x) satisfies A(x) = C(x) / (1 - x*A(x))^4, where C(x) is the g.f. of A000108.

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A127631 Square of Riordan array (1, x*c(x)) where c(x) is the g.f. of A000108.

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A368935 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x+x^3) ).

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A368936 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x+x^4) ).

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A368937 Expansion of (1/x) * Series_Reversion( x * (1-x) * (1-x+x^5) ).

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A381828 Expansion of ( (1/x) * Series_Reversion( x * ((1-x) * (1-x+x^2))^2 ) )^(1/2).

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