A026375 -id:A026375 - cp's OEIS frontend

A383120 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k).

1, 2, 11, 139, 2885, 82381, 2979565, 130203494, 6664589321, 390857822425, 25832193906761, 1899273577364197, 153741850998047053, 13585520026454056279, 1301210398133681268381, 134270617908678099820891, 14849785991790603714043921, 1752283118795349858851381297

Offset: 0

Ilya Gutkovskiy, Apr 17 2025

nonn

Cf. A001850, A014062, A026375, A188686, A226391, A359643, A378327, A383121.

Table[Sum[Binomial[n, k] Binomial[n k, k], {k, 0, n}], {n, 0, 17}]

a(n) = sum(k=0, n, binomial(n,k) * binomial(n*k,k)); \\ Michel Marcus, Apr 17 2025

a(n) = [x^n] ((1 + x)^n + x)^n.

a(n) ~ exp(n + exp(-1) - 1/2) * n^n / sqrt(2*Pi*n). - Vaclav Kotesovec, Apr 17 2025

A126177 Triangle read by rows: T(n,k) is number of hex trees with n edges and k leaves (n >= 1, 1 <= k <= 1 + floor(n/2)).

Original entry on oeis.org

3, 9, 1, 27, 9, 81, 54, 2, 243, 270, 30, 729, 1215, 270, 5, 2187, 5103, 1890, 105, 6561, 20412, 11340, 1260, 14, 19683, 78732, 61236, 11340, 378, 59049, 295245, 306180, 85050, 5670, 42, 177147, 1082565, 1443420, 561330, 62370, 1386, 531441, 3897234

Offset: 1

Emeric Deutsch, Dec 19 2006

A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a middle child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read paper).
Also number of hex trees with n edges and k-1 nodes of outdegree 2.
Row n has 1 + floor(n/2) terms.
Sum of terms in row n = A002212(n+1).
T(n,1) = 3^n (A000244).
T(n,2) = A027472(n+1).
Sum_{k=1..1+floor(n/2)} k*T(n,k) = A026375(n).

			Triangle starts:
    3;
    9,   1;
   27,   9;
   81,  54,   2;
  243, 270,  30;

F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.

Cf. A002212, A000244, A027472, A026375.

T:=(n,k)->3^(n-2*k+2)*binomial(2*k-2,k-1)*binomial(n,2*k-2)/k: for n from 1 to 13 do seq(T(n,k),k=1..1+floor(n/2)) od; # yields sequence in triangular form

T(n,k) = 3^(n-2k+2)*binomial(2k-2,k-1)*binomial(n,2k-2)/k. Proof: There are Catalan(k-1) full binary trees with k leaves. Each of them has 2k-2 edges. Additional n-2k+2 edges can be inserted as paths at the existing 2k-1 vertices in 3^(n-2k+2)*binomial(n,2k-2) ways.

G.f.: G=G(t,z) satisfies z^2*G^2-(1-3z-2tz^2)G+tz(3+tz)=0.

A293491 a(n) = n! * [x^n] exp((n+2)x)BesselI(0,2*x).

Original entry on oeis.org

1, 3, 18, 155, 1734, 23877, 390804, 7417377, 160256070, 3885021569, 104465601756, 3086353547433, 99399100528924, 3466411543407555, 130151205663179112, 5235127829223881895, 224609180728848273990, 10239557195235638377449, 494317596005491398892620, 25192788307121307053168673

Offset: 0

Ilya Gutkovskiy, Oct 10 2017

nonn

The n-th term of the n-th binomial transform of A000984.

N. J. A. Sloane, Transforms

Cf. A000984, A026375, A081671, A098409, A098410, A104454, A186925, A292627, A292632.

Table[n! SeriesCoefficient[Exp[(n + 2) x] BesselI[0, 2 x], {x, 0, n}], {n, 0, 19}]
Table[SeriesCoefficient[1/Sqrt[(1 - n x) (1 - (n + 4) x)], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[Binomial[n, k] Binomial[2 k, k] n^(n - k), {k, 0, n}], {n, 1, 19}]]
Table[(n + 2)^n HypergeometricPFQ[{1/2 - n/2, -n/2}, {1}, 4/(2 + n)^2], {n, 0, 19}]

a(n) = [x^n] 1/sqrt((1 - n*x)*(1 - (n + 4)*x)).
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)*n^(n-k).
a(n) ~ exp(2) * BesselI(0,2) * n^n. - Vaclav Kotesovec, Oct 16 2017

A302180 Number of 3D walks of type aad.

Original entry on oeis.org

1, 1, 3, 7, 23, 71, 251, 883, 3305, 12505, 48895, 193755, 783355, 3205931, 13302329, 55764413, 236174933, 1008773269, 4343533967, 18834033443, 82201462251, 360883031291, 1592993944723, 7066748314147, 31493800133173, 140953938878821, 633354801073571, 2856369029213263

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.
Number of 3D walks of length n in the first octant using steps (1, 1, 0), (1, -1, 0), (1, 0, 1), (1, 0, -1) and (1, 0, 0) that start at the origin and end at (n, 0, 0). The analogous problem in 2D is given by the Motzkin numbers A001006. - Farzan Byramji, Mar 06 2021
Inverse binomial transform of A145867 (Number of 3D walks of type aae). - Mélika Tebni, Nov 05 2024

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A001006, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

M := n-> add(binomial(n, 2*k)*binomial(2*k, k)/(k+1), k = 0 .. iquo(n,2)): # Motzkin numbers
A302180 := n-> add((-1)^(n-k)*binomial(n, k)*add(binomial(k, j)*M(j)*M(k-j), j=0..k), k=0..n):  seq(A302180(n), n = 0 .. 26); # Mélika Tebni, Nov 05 2024

a(14)-a(26) from Farzan Byramji, Mar 06 2021

A302182 Number of 3D walks of type abc.

Original entry on oeis.org

1, 1, 5, 12, 62, 200, 1065, 3990, 21714, 89082, 492366, 2147376, 12004740, 54718092, 308559537, 1454116950, 8255788970, 39935276810, 227976044010, 1126178350440, 6457854821340, 32456552441040, 186814834574550, 952569927106980, 5500292590186380, 28391993275117500

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

Cf. A018224, A126120, A135394, A302181.

from math import comb as binomial
def row(n: int) -> list[int]:
    return sum(binomial(n, k)*binomial(k, k//2)//(k//2+1)*((k+1) %2)*binomial(n-k, (n-k)//2)**2 for k in range(n+1))
for n in range(26): print(row(n)) # Mélika Tebni, Nov 27 2024

From Mélika Tebni, Nov 27 2024: (Start)
a(n) = Sum_{k=0..n} binomial(n, k)*A126120(k)*A018224(n-k).
a(2*n+1) = A135394(n) / (2*n+2).
a(2*n) = A302181(n). (End)

a(13)-a(25) from Mélika Tebni, Nov 27 2024

A302184 Number of 3D walks of type abe.

Original entry on oeis.org

1, 2, 7, 26, 108, 472, 2159, 10194, 49396, 244328, 1229308, 6273896, 32410096, 169181664, 891181607, 4731912082, 25302648644, 136150941064, 736747902236, 4007011320808, 21893702201648, 120125750018656, 661630546993116, 3656966382542984, 20278320788680912, 112782556853239712

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A126120, A138547, A145847, A145867, A150500, A202814.

a := n -> 2*add(binomial(n, k)*binomial(k, k/2)*binomial(2*(n-k), n-k)/(k+2), k = 0..n, 2): seq(a(n), n = 0..25);  # Peter Luschny, Nov 30 2024

from math import comb as binomial
def a(n: int):
    return sum(binomial(n, k)*binomial(k, k//2)//(k//2+1)*((k+1) %2)*binomial(2*(n-k), n-k) for k in range(n+1))
print([a(n) for n in range(26)]) # Mélika Tebni, Nov 30 2024

a(n) = Sum_{k=0..n} binomial(n, k)*A126120(k)*A000984(n-k). - Mélika Tebni, Nov 30 2024

a(12)-a(25) from Mélika Tebni, Nov 30 2024

A349001 The number of Lyndon words of size n from an alphabet of 5 letters and 1st and 2nd letter of the alphabet with equal frequency in the words.

Original entry on oeis.org

1, 3, 4, 14, 46, 174, 656, 2640, 10790, 45340, 193600, 839820, 3686424, 16353924, 73187456, 330052646, 1498335650, 6841899606, 31404443032, 144814450188, 670552118244, 3116578216310, 14534401932712, 67992210407514, 318969964124256, 1500268062754830

Offset: 0

R. J. Mathar, Nov 05 2021

nonn

Counts a subset of the Lyndon words in A001692. Here there is no requirement of how often the 3rd to 5th letter of the alphabet are in the admitted word, only on the frequency of the 1st and 2nd letter of the alphabet.

Let T(n,k,M) be the number of words of length n drawn from an alphabet of size M where the first k letters of the alphabet appear with the same frequency f in each word. Then T(n,k,M) = Sum_{f=0..floor(n/k)} (M-k)^(n-f*k) * Product_{i=0..k-1} binomial(n-i*f,f) and T(n,2,5) = A026375(n), T(n,3,6) = A294035(n), T(n,2,6) = A081671(n). Removing the words with cycles by the inclusion-exclusion principle by a Mobius Transform gives words of length n of that type without cycles and division through n the Lyndon words of that type. - R. J. Mathar, Nov 07 2021

			Examples for the alphabet {0,1,2,3,4}:
a(0)=1 counts (), the empty word.
a(3)=14 counts (021) (031) (041) (012) (013) (223) (233) (243) (014) (224) (234) (334) (244) (344), words of length 3 where the letters 0 and the 1 occur both either not or once.
a(4)=46 counts (0011) (0221) (0321) (0421) (0231) (0331) (0431) (0241) (0341) (0441) (0212) (0312) (0412) (0122) (0132) (0142) (0213) (0313) (0413) (0123) (2223) (0133) (2233) (2333) (2433) (0143) (2243) (2343) (2443) (0214) (0314) (0414) (0124) (2224) (2324) (0134) (2234) (2334) (3334) (2434) (0144) (2244) (2344) (3344) (2444) (3444).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Lyndon words

Cf. A022553 (alphabet of 2 letters), A290277 (of 3 letters), A060165 (of 4 letters), A026375.

a(n) = if(n>0, sumdiv(n, d, moebius(n/d)*sum(k=0, d, binomial(d,k)*binomial(2*k,k)))/n, n==0) \\ Andrew Howroyd, Jan 14 2023

n*a(n) = Sum_{d|n} mu(d)*A026375(n/d) where mu = A008683.

Terms a(16) and beyond from Andrew Howroyd, Jan 14 2023

A370285 Coefficient of x^n in the expansion of ( (1+x)^2 + x^3 )^n.

Original entry on oeis.org

1, 2, 6, 23, 94, 392, 1659, 7107, 30734, 133880, 586576, 2582142, 11411371, 50597900, 224986467, 1002867878, 4479814606, 20049099908, 89878609344, 403521966942, 1814102538624, 8165526187128, 36794746597494, 165968135843522, 749314496125451, 3385881647958442

Offset: 0

Seiichi Manyama, Feb 14 2024

nonn

Cf. A006139, A026375, A369212.

Similar to A082759.

a := n -> binomial(2*n, n) * hypergeom([(1-n)/3, (2-n)/3, -n/3], [1/2-n, n+1], 27/4):
seq(simplify(a(n)), n = 0..25);  # Peter Luschny, Jan 04 2025

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

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

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / ((1+x)^2 + x^3) ). See A369212.

A110221 Triangle read by rows: T(n,k) (0<=k<=floor(n/2)) is the number of Delannoy paths of length n, having k ED's.

Original entry on oeis.org

1, 3, 11, 2, 45, 18, 195, 120, 6, 873, 720, 90, 3989, 4110, 870, 20, 18483, 22806, 6930, 420, 86515, 124264, 49560, 5320, 70, 408105, 668520, 331128, 52920, 1890, 1936881, 3562830, 2111760, 456120, 29610, 252, 9238023, 18850590, 13020480, 3575880

Offset: 0

Emeric Deutsch, Jul 20 2005

A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).

Row n has 1+floor(n/2) terms. Row sums are the central Delannoy numbers (A001850). Column 0 yields A026375. Sum(k*T(n,k),k=0..floor(n/2))=2*A002695(n).

			T(2,1)=2 because we have NED and EDN.
Triangle begins:
1;
3;
11,2;
45,18;
195,120,6;

Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Cf. A001850, A026375, A002695.

R:=(1-z-sqrt(1-6*z+5*z^2-4*z^2*t))/2/z/(1-z+t*z): G:=1/(1-z-2*t*z^2*R-2*z*R+2*z^2*R): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 12 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields sequence in triangular form

G.f.: 1/(1-z-2tz^2*R-2zR+2z^2*R), where R=[1-z-sqrt(1-6z+5z^2-4tz^2)]/[2z(1-z+tz)].

A168216 Riordan array (1/(1-x),xc(x)/(1-xc(x))) where c(x)is the g.f. of A000108.It factorizes as A007318*A106566.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 5, 1, 1, 23, 19, 7, 1, 1, 74, 69, 34, 9, 1, 1, 262, 256, 147, 53, 11, 1, 1, 993, 986, 615, 265, 76, 13, 1, 1, 3943, 3935, 2571, 1235, 431, 103, 15, 1, 1, 16178, 16169, 10862, 5591, 2216, 653, 134, 17, 1

Offset: 0

Philippe Deléham, Nov 20 2009

Inverse is Riordan array (1/(1+x-x^2),x(1-x)/(1+x-x^2)) = [ -1,-1,1,0,0,0,0,0,...]DELTA[1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. Unsigned version of A091698.

			Triangle begins : 1 ; 1,1 ; 1,3,1 ; 1,8,5,1 ; 1,23,19,7,1 ; ...

Cf. A000108, A007318, A106566.

Sum_{k, 0<=k<=n}T(n,k)*x^k = A000012(n), A007317(n+1), A026375(n) for x = 0, 1, 2 respectively.

cp's OEIS Frontend

A383120 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k).

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A126177 Triangle read by rows: T(n,k) is number of hex trees with n edges and k leaves (n >= 1, 1 <= k <= 1 + floor(n/2)).

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A293491 a(n) = n! * [x^n] exp((n+2)*x)*BesselI(0,2*x).

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A302180 Number of 3D walks of type aad.

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A302182 Number of 3D walks of type abc.

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A302184 Number of 3D walks of type abe.

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A349001 The number of Lyndon words of size n from an alphabet of 5 letters and 1st and 2nd letter of the alphabet with equal frequency in the words.

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A370285 Coefficient of x^n in the expansion of ( (1+x)^2 + x^3 )^n.

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A293491 a(n) = n! * [x^n] exp((n+2)x)BesselI(0,2*x).