A006013 -id:A006013 - cp's OEIS frontend

A286785 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

1, 2, 5, 2, 14, 14, 2, 42, 72, 27, 2, 132, 330, 220, 44, 2, 429, 1430, 1430, 520, 65, 2, 1430, 6006, 8190, 4550, 1050, 90, 2, 4862, 24752, 43316, 33320, 11900, 1904, 119, 2, 16796, 100776, 217056, 217056, 108528, 27132, 3192, 152, 2, 58786, 406980, 1046520, 1302336, 854658, 301644, 55860, 5040, 189, 2, 208012, 1634380, 4903140, 7354710, 6056820, 2826516, 743820, 106260, 7590, 230, 2

Offset: 0

Gheorghe Coserea, May 15 2017

Row n>0 contains n terms.

T(n,k) is the number of Feynman's diagrams with k fermionic loops in the order n of the perturbative expansion in dimension zero for the GW approximation of the polarization function in a many-body theory of fermions with two-body interaction (see Molinari link).

			A(x;t) = 1 + 2*x + (5 + 2*t)*x^2 + (14 + 14*t + 2*t^2)*x^3 + ...
Triangle starts:
   n\k |     0       1       2       3       4      5     6    7  8
  -----+-----------------------------------------------------------
   0   |     1;
   1   |     2;
   2   |     5,      2;
   3   |    14,     14,      2;
   4   |    42,     72,     27,      2;
   5   |   132,    330,    220,     44,      2;
   6   |   429,   1430,   1430,    520,     65,     2;
   7   |  1430,   6006,   8190,   4550,   1050,    90,    2;
   8   |  4862,  24752,  43316,  33320,  11900,  1904,  119,   2;
   9   | 16796, 100776, 217056, 217056, 108528, 27132, 3192, 152, 2;

Gheorghe Coserea, Rows n = 0..123, flattened

Cf. A000108, A286781, A286782, A286783.

T(n,k):=(binomial(n-1,k)*binomial(2*(n+1),n-k))/(n+1); /* Vladimir Kruchinin, Jan 14 2022 */

A286784_ser(N,t='t) = my(x='x+O('x^N)); serreverse(Ser(x*(1-x)^2/(1+(t-1)*x)))/x;
A286785_ser(N,t='t) = 1/(1-x*A286784_ser(N,t))^2;
concat(apply(p->Vecrev(p), Vec(A286785_ser(12))))

y(x;t) = Sum_{n>=0} P_n(t)*x^n = 1/(1-x*s)^2, where s(x;t) = A286784(x;t) and P_n(t) = Sum_{k=0..n-1} T(n,k)*t^k for n>0.
A000108(n+1) = T(n,0), A002058(n+3) = T(n,1), A014106(n-1) = T(n,n-2), A006013(n) = P_n(1), A211789(n+1) = P_n(2).
T(n,k) = C(n-1,k)*C(2*n+2,n-k)/(n+1). - Vladimir Kruchinin, Jan 14 2022

A111160 G.f.: C - Z; where C is the g.f. for the Catalan numbers (A000108) and Z is the g.f. for A055113 with offset 0.

Original entry on oeis.org

0, 1, 1, 4, 9, 31, 91, 309, 1009, 3481, 11956, 42065, 148655, 532039, 1915369, 6950452, 25357233, 93034813, 342888250, 1269246437, 4715945712, 17583623988, 65766726906, 246694006971, 927801717255, 3497918129001, 13217196871126, 50046561077947

Offset: 0

N. J. A. Sloane, Oct 22 2005

nonn

Expressible in terms of ballot numbers.

Number of positive walks with n steps {-2,-1,1,2} starting at the origin, ending at altitude 2, and staying strictly above the x-axis. - David Nguyen, Dec 16 2016

T. D. Noe, Table of n, a(n) for n=0..200
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.
Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022).
D. G. Rogers, Comments on A111160, A055113 and A006013

Cf. A000108, A055113, A006013, A187430, A276901, A306668.

I:=[1,1,4]; [0] cat [n le 3 select I[n] else (n*(115*n^3 - 344*n^2 + 299*n - 82)*Self(n-1) + 4*(2*n-3)*(5*n^3 + 27*n^2 - 74*n + 30)*Self(n-2) - 36*(n-2)*(2*n-5)*(2*n-3)*(5*n-3)*Self(n-3))/(2*n*(n+1)*(2*n+1)*(5*n-8)): n in [1..30]]; // Vincenzo Librandi, Oct 06 2015

a := n -> (-1)^(n+1)*binomial(2*n+1,n)*hypergeom([-n-1,n/2+1/2,n/2],[n,n+1],4)/ (2*n+1);
[0, op([seq(round(evalf(a(n),32)), n=1..27)])]; # Peter Luschny, Oct 06 2015

CoefficientList[ Series[ -((-3 + Sqrt[1 - 4*x] + Sqrt[2]*Sqrt[1 + Sqrt[1 - 4x] + 6x])/(4x)), {x, 0, 10}], x] (* Robert G. Wilson v *)

a(n) = if(n==0, 0, sum(k=0, (n+1)/2, binomial(n-k,n-2*k+1)*binomial(2*n+1,k))/(2*n+1)); \\ Altug Alkan, Oct 05 2015

Let C := (1 - sqrt(1 - 4*x)) / (2*x), Z := (- 1/4 - (1/4)*(1 - 4*x)^(1/2) + (1/4)*(2 + 2*(1 - 4*x)^(1/2) + 12*x)^(1/2))/x; g.f. is W := C - Z.
G.f.: -((-3 + sqrt(1 - 4x) + sqrt(2)*sqrt(1 + sqrt(1 - 4x) + 6x))/(4x)).
a(n) = sum(j=0..n+1, binomial(n+2*j-1,j)*(-1)^(n+j+1)*binomial(2*n+1,j+n))/(2*n+1). [Vladimir Kruchinin, Feb 15 2013]
a(n) ~ (1+1/sqrt(5))*2^(2*n-1)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013
Recurrence: 2*n*(n+1)*(2*n+1)*(5*n-8)*a(n) = n*(115*n^3 - 344*n^2 + 299*n - 82)*a(n-1) + 4*(2*n-3)*(5*n^3 + 27*n^2 - 74*n + 30)*a(n-2) - 36*(n-2)*(2*n-5)*(2*n-3)*(5*n-3)*a(n-3). - Vaclav Kotesovec, Aug 13 2013
a(n) = Sum_{j=0..(n+1)/2}(binomial(n-j,n-2*j+1)*binomial(2*n+1,j))/(2*n+1). - Vladimir Kruchinin, Oct 05 2015
a(n) = (-1)^(n+1)*C(2*n+1,n)*hypergeom([-n-1,n/2+1/2,n/2],[n,n+1],4)/(2*n+1) for n>0. - Peter Luschny, Oct 06 2015

A126042 Expansion of f(x^3)/(1-x*f(x^3)), where f(x) is the g.f. of A001764, whose n-th term is binomial(3n,n)/(2n+1).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 8, 13, 19, 38, 64, 98, 196, 337, 531, 1062, 1851, 2974, 5948, 10468, 17060, 34120, 60488, 99658, 199316, 355369, 590563, 1181126, 2115577, 3540464, 7080928, 12731141, 21430267, 42860534, 77306428, 130771376, 261542752, 473018396, 803538100

Offset: 0

Paul Barry, Dec 16 2006

Row sums of number triangle A111373.
Interleaves T(3n,2n), T(3n+1,2n+1) and T(3n+2,2n+2) for T(n,k) = A047089(n,k).
One step forward and two steps back: number of nonnegative walks of n steps where the steps are size 1 forwards and size 2 backwards. - David Scambler, Mar 15 2011
Brown's criterion ensures that the sequence is complete (see formulae). - Vladimir M. Zarubin, Aug 05 2019
Number of ordered trees with n+1 edges, having nonroot nodes of outdegree 0 or 3. - Emanuele Munarini, Jun 20 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, The Triple Riordan Group, arXiv:2412.05461 [math.CO], 2024. See pp. 7, 10.
Eric Weisstein's World of Mathematics, Brown's Criterion.

Cf. A047089, A111373.

[n lt 3 select 1 else Binomial(n, Floor(n/3)) - (&+[Binomial(n,j): j in [0..Floor(n/3)-1]]): n in [0..40]]; // G. C. Greubel, Jul 30 2022

a:= proc(n) option remember; `if`(n<4, [1$3, 2][n+1], (a(n-1)*
       2*(20*n^4-14*n^3-31*n^2-n+8)-6*(3*n-1)*(5*n-6)*a(n-2)
      +9*(n-2)*(15*n^3-48*n^2+15*n+14)*a(n-3)-54*(n-2)*(n-3)*
      (5*n^2-n-2)*a(n-4))/(2*(2*n+1)*(n+1)*(5*n^2-11*n+4)))
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Sep 07 2022

Table[Binomial[n, Floor[n/3]] -Sum[Binomial[n,i], {i,0,Floor[n/3] -1}], {n,0,40}] (* David Callan, Oct 26 2017 *)
a[n_] := Binomial[n, Floor[n/3]] (1 + Hypergeometric2F1[1, -n + Floor[n/3], 1 + Floor[n/3], -1]) - 2^n; Table[a[n], {n, 0, 38}] (* Peter Luschny, Jun 20 2024 *)

{a(n)=polcoeff((1/x)*serreverse(x*(1+x)^2/((1+x)^3+x^3+x*O(x^n))),n)}

n=30;
{a0=1;a1=1;a2=1;for(k=1, n/3,print1(a0,", ",a1,", ",a2,", ");
a0=2*a2;a1=2*a0-binomial(3*k,k)/(2*k+1);a2=2*a1-binomial(3*k+1,k)/(k+1))
} \\ Vladimir M. Zarubin, Aug 05 2019

[binomial(n, (n//3)) - sum(binomial(n,j) for j in (0..(n//3)-1)) for n in (0..40)] # G. C. Greubel, Jul 30 2022

a(n) = Sum_{k=0..n} binomial(3*floor((n+2k)/3) - 2k, floor((n+2k)/3)-k)*(k+1)/(2*floor((n+2k)/3) - k + 1)(2*cos(2*Pi*(n-k)/3) + 1)/3.
G.f.: (1/x)*Series_Reversion( x*(1+x)^2/((1+x)^3+x^3) ). - Paul D. Hanna, Mar 15 2011
From Vladimir M. Zarubin, Aug 05 2019: (Start)
a(0) = 1, a(1) = 1, a(2) = 1 and for k>0
a(3*k) = 2*a(3*k-1),
a(3*k+1) = 2*a(3*k) - binomial(3*k,k)/(2*k+1),
a(3*k+2) = 2*a(3*k+1) - binomial(3*k+1,k)/(k+1),
where binomial(3*k,k)/(2*k+1) = A001764(k)
and binomial(3*k+1,k)/(k+1) = A006013(k). (End)
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (n-3*k+1) * binomial(n+1,k). - Seiichi Manyama, Jan 27 2024

A329058 2-parking triangle T(r, i, 2) read by rows: T(r, i, k) = (r + 1)^(i-1)binomial(k(r + 1) + r - i - 1, r - i) with k = 2 and 0 <= i <= r.

Original entry on oeis.org

1, 2, 1, 7, 6, 3, 30, 36, 32, 16, 143, 220, 275, 250, 125, 728, 1365, 2184, 2808, 2592, 1296, 3876, 8568, 16660, 27440, 36015, 33614, 16807, 21318, 54264, 124032, 248064, 417792, 557056, 524288, 262144, 120175, 346104, 908523, 2133054, 4363065, 7479540, 10097379, 9565938, 4782969

Offset: 0

Stefano Spezia, Nov 02 2019

The k-parking numbers interpolate between the generalized Fuss-Catalan numbers and the number of parking functions (see Yip).

			r/i|   0   1   2   3   4
————————————————————————
0  |   1
1  |   2   1
2  |   7   6   3
3  |  30  36  32  16
4  | 143 220 275 250 125

Stefano Spezia, First 151 rows of the triangle, flattened
Martha Yip, A Fuss-Catalan variation of the caracol flow polytope, arXiv:1910.10060 [math.CO], 2019.

Cf. A000108, A000272, A006013, A007318, A329057, A329059, A329060, A329113 (row sums).

T[r_, i_,k_] := (r + 1)^(i-1)*Binomial[k*(r + 1) + r - i - 1, r - i]; Flatten[Table[T[r,i,2],{r,0,9},{i,0,r}]]

T(r, i, k) = (r + 1)^(i-1)*binomial(k*(r + 1) + r - i - 1, r - i).
T(r, 0, 2) = A006013(r).
T(r, r, 2) = A000272(r + 1).

A361306 Expansion of A(x) satisfying A(x) = x + A(x)^2*(1 + A(x))^4.

Original entry on oeis.org

1, 1, 6, 31, 186, 1191, 7972, 55164, 391322, 2830751, 20801826, 154853413, 1165316224, 8850372878, 67750780816, 522218420336, 4049564739054, 31570368061361, 247293510244174, 1945331619223591, 15361731119713506, 121729460653957980, 967664450692965300

Offset: 1

Paul D. Hanna, Mar 08 2023

nonn

			G.f.: A(x) = x + x^2 + 6*x^3 + 31*x^4 + 186*x^5 + 1191*x^6 + 7972*x^7 + 55164*x^8 + 391322*x^9 + ...
such that sqrt(A(x) - x) = A(x)*(1 + A(x))^2.
A(x)*(1 + A(x))^2 = x + 3*x^2 + 11*x^3 + 60*x^4 + 355*x^5 + 2261*x^6 + 15094*x^7 + 104208*x^8 + ...
A(x)*(1 + A(x))^2 = Series_Reversion( -x^2 + Sum_{n>=1} (-1)^(n-1) * binomial(3*n-2,n-1)*x^n/n ).

Paul D. Hanna, Table of n, a(n) for n = 1..500

Cf. A361304, A361305, A214372, A006013.

{a(n)=polcoeff(serreverse(x - x^2*(1+x)^4 +x*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^(2*m)*(1+x+x*O(x^n))^(4*m)/m!)); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(2*m-1)*(1+x+x*O(x^n))^(4*m)/m!))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following.
(1) A(x) = Series_Reversion( x - x^2*(1+x)^4 ).
(2) A(x) = x + A(x)^2 * (1 + A(x))^4.
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n) * (1+x)^(4*n) / n!.
(4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1+x)^(4*n) / n! ).
(5) A(x) = x + Series_Reversion( Series_Reversion( x*(1+x)^2 ) - x^2 )^2.
(6) A(x) = x + Series_Reversion( -x^2 + Sum_{n>=1} (-1)^(n-1) * binomial(3*n-2,n-1) * x^n/n )^2.
From Vaclav Kotesovec, Mar 09 2023: (Start)
Recurrence: 3381*(n-4)*(n-3)*(n-2)*(n-1)*n*(4485934293448*n^5 - 88905588075732*n^4 + 698950092208066*n^3 - 2724285958475163*n^2 + 5263801532363671*n - 4032831805999290)*a(n) = 2*(n-4)*(n-3)*(n-2)*(n-1)*(33204885640102096*n^6 - 707886491396721408*n^5 + 6160367858867908768*n^4 - 27918165429184721124*n^3 + 69150795811214975011*n^2 - 88077097294043237943*n + 44480953779348451050)*a(n-1) + 8*(n-4)*(n-3)*(n-2)*(52772531028122272*n^7 - 1256975462235400336*n^6 + 12611049851568548176*n^5 - 69004162305753446968*n^4 + 222104765912229832762*n^3 - 419924105934755620321*n^2 + 431120275047208552290*n - 185089750933520270250)*a(n-2) + 48*(n-4)*(n-3)*(17647665510424432*n^8 - 482112074818112928*n^7 + 5693971809001104840*n^6 - 37956706633792772384*n^5 + 156126872715173363823*n^4 - 405548028261835673882*n^3 + 649232078072133939050*n^2 - 585187986606994739801*n + 227161430445970883100)*a(n-3) + 32*(n-4)*(21945190563547616*n^9 - 698268423629052336*n^8 + 9788485232517982416*n^7 - 79313303231764021176*n^6 + 409187506797434806734*n^5 - 1393249646753024170299*n^4 + 3129189249705937191544*n^3 - 4467594298222926610959*n^2 + 3676695031470911619960*n - 1327813620065788842000)*a(n-4) + 72*(2*n - 7)*(3*n - 14)*(3*n - 13)*(6*n - 31)*(6*n - 29)*(4485934293448*n^5 - 66475916608492*n^4 + 388187082839618*n^3 - 1116009867370877*n^2 + 1578887211201855*n - 878785793685000)*a(n-5).
a(n) ~ 1/(2 * (1 + s) * sqrt(Pi*(1 + 10*s + 15*s^2)) * n^(3/2) * r^(n - 1/2)), where r = 0.1176087332021218420455915375218722861407778043565... and s = 0.1894485384658193296593809633217117092941452563863... are real roots of the system of equations r + s^2 * (1+s)^4 = s, 2*s*(1+s)^3 * (1+3*s) = 1. (End)
a(n+1) = Sum_{k=0..n} binomial(n+k+1,k) * binomial(4*k,n-k)/(n+k+1). - Seiichi Manyama, Aug 24 2023

A366326 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^2).

Original entry on oeis.org

1, 2, -3, 14, -78, 479, -3131, 21372, -150588, 1087057, -7998295, 59763129, -452257495, 3459109408, -26697940390, 207672518808, -1626400971710, 12813379464399, -101482102525511, 807524595076284, -6452856224076654, 51760509258982478, -416620859045829372

Offset: 0

Seiichi Manyama, Oct 07 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..1067

Cf. A073157, A364336, A364337, A364338, A364339, A366325, A366327, A366328.

Cf. A006013, A364393.

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

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

A369616 Expansion of (1/x) * Series_Reversion( x / (1/(1-x)^2 + x) ).

Original entry on oeis.org

1, 3, 12, 58, 314, 1824, 11107, 69955, 451918, 2977834, 19936332, 135225006, 927267595, 6417580459, 44770275705, 314489676679, 2222549047262, 15791353483602, 112734135824404, 808247711066688, 5817056710700424, 42012120642574732, 304384379305912686

Offset: 0

Seiichi Manyama, Jan 27 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for reversions of series

Cf. A007317, A369617, A370844.

Cf. A006013, A274734.

A369616 := proc(n)
    add(binomial(n+1,k) * binomial(3*n-3*k+1,n-k),k=0..n) ;
    %/(n+1) ;
end proc;
seq(A369616(n),n=0..70) ; # R. J. Mathar, Jan 28 2024

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

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

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

D-finite with recurrence 2*(n+1)*(2*n+1)*a(n) +3*(-13*n^2+1)*a(n-1) +33*(2*n-1)*(n-1)*a(n-2) -31*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jan 28 2024

A370474 G.f. A(x) satisfies A(x) = 1 + x * A(x)^(3/2) * (1 + A(x)^(3/2)).

Original entry on oeis.org

1, 2, 9, 54, 372, 2778, 21873, 178786, 1502649, 12904524, 112741664, 998871030, 8953443276, 81047485148, 739846170864, 6803054508702, 62954736555836, 585850907166084, 5479077065774682, 51470699845616004, 485456696541512442, 4595280949098247422

Offset: 0

Seiichi Manyama, Mar 31 2024

nonn

Cf. A006013, A370475.
Cf. A262441, A366400.
Cf. A271469.

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

a(n, r=2, s=-1, t=5, u=3) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 12 2024

a(n) = Sum{k=0..n} binomial(n,k) * binomial(3*n/2+3*k/2+1,n)/(3*n/2+3*k/2+1).
From Seiichi Manyama, Dec 12 2024: (Start)
G.f. A(x) satisfies:
(1) A(x) = ( 1 + x*A(x)^(5/2)/(1 + x*A(x)^(3/2)) )^2.
(2) A(x) = 1/( 1 - x*A(x)^2/(1 + x*A(x)^(3/2)) )^2.
(3) A(x) = B(x)^2 where B(x) is the g.f. of A271469.
If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r). (End)

A371494 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1+x))^2.

Original entry on oeis.org

1, 2, 5, 18, 72, 310, 1399, 6532, 31287, 152876, 759034, 3818410, 19420713, 99697784, 515909606, 2688267462, 14093211259, 74281217492, 393389969722, 2092312452404, 11171325560120, 59854910468196, 321717833732186, 1734250394445622, 9373581927760595

Offset: 0

Seiichi Manyama, Mar 25 2024

nonn

Cf. A001006, A371495, A371496.

Cf. A006013.

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

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

A377503 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^2.

Original entry on oeis.org

1, 2, 18, 270, 5936, 173330, 6335772, 278724362, 14350790064, 847007698338, 56397332340020, 4182866692785242, 342022887565717800, 30570009715185100082, 2965368922693150575084, 310276298423966343555690, 34834957115496822249510752, 4177193847524372747798263106

Offset: 0

Seiichi Manyama, Oct 30 2024

nonn

Cf. A295238, A377504, A377528.

Cf. A006013, A364983.

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

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364983.

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

cp's OEIS Frontend

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