A308523 -id:A308523 - cp's OEIS frontend

A036829 a(n) = Sum_{k=0..n-1} C(3k,k)C(3n-3k-2,n-k-1).

0, 1, 7, 48, 327, 2221, 15060, 102012, 690519, 4671819, 31596447, 213633696, 1444131108, 9760401756, 65957919496, 445671648228, 3011064814455, 20341769686311, 137412453018933, 928188965638464, 6269358748632207, 42343731580741821

Offset: 0

N. J. A. Sloane

nonn

M. Petkovsek et al., A=B, Peters, 1996, p. 97.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A006256.
Cf. A001764, A006013, A007318, A036917, A062236.
Cf. A000302, A308523, A386367, A386368.

a036829 n = sum $ map
   (\k -> (a007318 (3*k) k) * (a007318 (3*n-3*k-2) (n-k-1))) [0..n-1]
-- Reinhard Zumkeller, May 24 2012

Table[Sum[Binomial[3k,k]Binomial[3n-3k-2,n-k-1],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Jan 10 2012 *)

G.f.: (g-g^2)/(3*g-1)^2 where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011
Recurrence: 8*(n-1)*(2*n-1)*a(n) = 6*(36*n^2-81*n+49)*a(n-1) - 81*(3*n-5)*(3*n-4)*a(n-2). - Vaclav Kotesovec, Nov 19 2012
a(n) ~ 3^(3*n-1)/2^(2*n+1). - Vaclav Kotesovec, Dec 29 2012
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/3) * log( Sum_{k>=0} binomial(3*k,k)*x^k ). - Seiichi Manyama, Jul 19 2025
G.f.: (g-1)/(3-2*g)^2 where g=1+x*g^3. - Seiichi Manyama, Jul 26 2025

A386612 a(n) = Sum_{k=0..n-1} binomial(4k+1,k) binomial(4n-4k,n-k-1).

Original entry on oeis.org

0, 1, 13, 142, 1464, 14689, 145154, 1420812, 13818784, 133793940, 1291073809, 12426782294, 119371355672, 1144851458526, 10965655515588, 104919037771224, 1002960800712720, 9580390527192940, 91453374122574372, 872513477065735768, 8320168165323802464, 79305962393873976417

Offset: 0

Seiichi Manyama, Jul 27 2025

nonn

Cf. A078995, A308523, A386565, A386611.

Cf. A006419, A386614, A386616, A386617.

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

G.f.: g^3 * (g-1)/(4-3*g)^2 where g=1+x*g^4.
G.f.: g/((1-g)^2 * (1-4*g)^2) where g*(1-g)^3 = x.
a(n) = Sum_{k=0..n-1} binomial(4*k+1+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n+2,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k+2,k).
D-finite with recurrence 13122*n*(3*n+2)*(3*n+1)*a(n) +81*(-124803*n^3+284553*n^2-210740*n+42140)*a(n-1) +24*(7476768*n^3-29253744*n^2+37920106*n-16562575)*a(n-2) +40960*(-26344*n^3+148032*n^2-282329*n+185874)*a(n-3) +55050240*(2*n-5)*(4*n-13)*(4*n-11)*a(n-4)=0. - R. J. Mathar, Aug 10 2025

A386368 a(n) = Sum_{k=0..n-1} binomial(6k,k) binomial(6n-6k-2,n-k-1).

Original entry on oeis.org

0, 1, 16, 246, 3736, 56421, 849432, 12763878, 191548464, 2871970110, 43031833656, 644432826478, 9646983339456, 144366433138955, 2159869510669320, 32306874783230556, 483151884326658144, 7224464127509984490, 108011596038055519680, 1614676987907480393940

Offset: 0

Seiichi Manyama, Jul 19 2025

nonn

			(1/6) * log( Sum_{k>=0} binomial(6*k,k)*x^k ) = x + 8*x^2 + 82*x^3 + 934*x^4 + 56421*x^5/5 + ...

Cf. A000302, A036829, A308523, A386367.
Cf. A079679, A386567, A386615, A386616.
Cf. A002295, A130564.

A386368 := proc(n::integer)
    add(binomial(6*k,k)*binomial(6*n-6*k-2,n-k-1),k=0..n-1) ;
end proc:
seq(A386368(n),n=0..80) ; # R. J. Mathar, Jul 30 2025

a(n) = sum(k=0, n-1, binomial(6*k, k)*binomial(6*n-6*k-2, n-k-1));

my(N=20, x='x+O('x^N), g=x*sum(k=0, N, binomial(6*k+4, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-6*g)^2))

G.f.: g*(1-g)/(1-6*g)^2 where g*(1-g)^5 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/6) * log( Sum_{k>=0} binomial(6*k,k)*x^k ).
G.f.: (g-1)/(6-5*g)^2 where g=1+x*g^6.
a(n) = Sum_{k=0..n-1} binomial(6*k-2+l,k) * binomial(6*n-6*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(6*n-1,k).
a(n) = Sum_{k=0..n-1} 6^(n-k-1) * binomial(5*n+k-1,k).
Conjecture D-finite with recurrence 48828125*(n-1)*(5*n-4)*(5*n-3) *(432862082629612805*n -769306661967834399) *(5*n-2)*(5*n-1)*a(n) +1125000*(-405245406115816219575000*n^6 +2613180799468910510392500*n^5 -7667164406968651479521250*n^4 +13834502135358262506660375*n^3 -16251583347734702117341345*n^2 +11251247074043948959380314*n -3395699069351241765495720)*a(n-1) +33592320*(142281690918326440537500*n^6 -1266424338521609272012500*n^5 +5236041263583271687953750*n^4 -12786608152035075786775875*n^3 +18838556229131595646260055*n^2 -15323925851720394901667853*n +5240681406952416812161236)*a(n-2) -53444359913472*(6*n-17) *(395547729523405*n -538181211711288)*(6*n-13) *(3*n-7)*(2*n-5) *(3*n-8)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

A386565 a(n) = Sum_{k=0..n-1} binomial(4k-1,k) binomial(4n-4k,n-k-1).

Original entry on oeis.org

0, 1, 11, 111, 1091, 10596, 102237, 982458, 9415539, 90063180, 860278156, 8208539351, 78258171957, 745595635084, 7099714918062, 67574576298276, 642927956583123, 6115089154367484, 58146652079312580, 552769690436583532, 5253812277363417836, 49925987913040522128

Offset: 0

Seiichi Manyama, Jul 26 2025

nonn

			(1/3) * log( Sum_{k>=0} binomial(4*k-1,k)*x^k ) = x + 11*x^2/2 + 37*x^3 + 1091*x^4/4 + 10596*x^5/5 + ...

Cf. A000346, A062236, A386566, A386567.

Cf. A002293, A006632, A308523.

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

my(N=30, x='x+O('x^N), g=sum(k=0, N, binomial(4*k, k)/(3*k+1)*x^k)); concat(0, Vec(g*(g-1)/(4-3*g)^2))

G.f.: g*(g-1)/(4-3*g)^2 where g=1+x*g^4.
G.f.: g/(1-4*g)^2 where g*(1-g)^3 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/3) * log( Sum_{k>=0} binomial(4*k-1,k)*x^k ).
a(n) = Sum_{k=0..n-1} binomial(4*k-1+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k,k).

A386367 a(n) = Sum_{k=0..n-1} binomial(5k,k) binomial(5n-5k-2,n-k-1).

Original entry on oeis.org

0, 1, 13, 163, 2021, 24930, 306655, 3765448, 46182101, 565939603, 6931070490, 84845250370, 1038235255415, 12700966517968, 155336699256808, 1899439862390640, 23222289820948405, 283872591297526505, 3469680960837171415, 42404345427419774621, 518193229118757697930

Offset: 0

Seiichi Manyama, Jul 19 2025

nonn

			(1/5) * log( Sum_{k>=0} binomial(5*k,k)*x^k ) = x + 13*x^2/2 + 163*x^3/3 + 2021*x^4/4 + 4986*x^5 + ...

Cf. A000302, A036829, A308523, A386368.
Cf. A079678, A386566, A386613, A386614.
Cf. A002294, A118971.

a(n) = sum(k=0, n-1, binomial(5*k, k)*binomial(5*n-5*k-2, n-k-1));

my(N=30, x='x+O('x^N), g=x*sum(k=0, N, binomial(5*k+3, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-5*g)^2))

G.f.: g*(1-g)/(1-5*g)^2 where g*(1-g)^4 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/5) * log( Sum_{k>=0} binomial(5*k,k)*x^k ).
G.f.: (g-1)/(5-4*g)^2 where g=1+x*g^5.
a(n) = Sum_{k=0..n-1} binomial(5*k-2+l,k) * binomial(5*n-5*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n-1,k).
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k-1,k).

A386611 a(n) = Sum_{k=0..n-1} binomial(4k,k) binomial(4n-4k,n-k-1).

Original entry on oeis.org

0, 1, 12, 126, 1268, 12513, 122148, 1184364, 11432100, 109997460, 1055891248, 10117633542, 96812495820, 925334377822, 8836315646616, 84317468847768, 804064275489924, 7663595943744876, 73009005101019792, 695263276434909976, 6618709687608909648, 62989317586872238689

Offset: 0

Seiichi Manyama, Jul 27 2025

nonn

Cf. A308523, A386565, A386612.

Cf. A008549, A386613, A386615.

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

G.f.: g^2 * (g-1)/(4-3*g)^2 where g=1+x*g^4.
G.f.: g/((1-g) * (1-4*g)^2) where g*(1-g)^3 = x.
a(n) = Sum_{k=0..n-1} binomial(4*k+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k+1,k).

A308524 Number of essentially 3-connected rooted toroidal maps with n edges.

Original entry on oeis.org

0, 0, 1, 2, 11, 40, 166, 658, 2647, 10592, 42446, 169972, 680670, 2725320, 10910992, 43678882, 174843151, 699839680, 2801078662, 11210671612, 44866276906, 179552951440, 718539964132, 2875389341332, 11506176209206, 46042099714240, 184234059839116, 737184620655368

Offset: 0

Nicolas Bonichon, Jun 05 2019

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Nicolas Bonichon, Éric Fusy, Benjamin Lévêque, A bijection for essentially 3-connected toroidal maps, arXiv:1907.04016 [math.CO], 2019.

Cf. A308523, A308526, A289208, A006422.

dev_A := 0; n := 20; dev_A := series(RootOf(A-x*(1+A)^2, A), x = 0, n+1);
seq(coeff(series(subs(A = dev_A, A^2*(1+A)/((1+2*A)*(1-A)^2*(1+3*A))), x, n+1), x, k), k = 0 .. n);
# second Maple program:
a:= proc(n) option remember; `if`(n<6, [0, 0, 1, 2, 11, 40][n+1],
     ((37*n^2-258*n+401)*a(n-1)-6*(2*n^2-25*n+88)*a(n-2)
      -48*(3*n^2-23*n+45)*a(n-3)-32*(n-4)*(2*n-7)*a(n-4))
      /((6*(n-1))*(n-5)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 07 2019

CoefficientList[Series[x*(1 + 8*x + (2*x - 1)*Sqrt[1 - 4*x])/(2*(2 + x)*(1 - 4*x)*(3 + 4*x)), {x, 0, 30}], x] (* Vaclav Kotesovec, Jun 25 2019 *)

G.f.: A^2*(1+A)/((1+2*A)*(1-A)^2*(1+3*A)) where A=x*(1+A)^2.
G.f.: x*(1 + 8*x + (2*x - 1)*sqrt(1 - 4*x))/(2*(2 + x)*(1 - 4*x)*(3 + 4*x)). - Vaclav Kotesovec, Jun 25 2019
a(n) ~ 2^(2*n - 3) / 3. - Vaclav Kotesovec, Jun 25 2019

A308526 Number of essentially 3-connected rooted toroidal maps with n vertices.

Original entry on oeis.org

0, 2, 42, 892, 18888, 399280, 8431776, 177936064, 3753206400, 79139040000, 1668268861952, 35160393493504, 740921108899840, 15611120289755136, 328889518650990592, 6928313584957702144, 145939409585973133312, 3073901537848967495680, 64741608434203590524928

Offset: 0

Nicolas Bonichon, Jun 05 2019

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..500
Nicolas Bonichon, Éric Fusy, Benjamin Lévêque, A bijection for essentially 3-connected toroidal maps, arXiv:1907.04016 [math.CO], 2019.

Cf. A308523, A308524, A289208, A006422.

dev_A := 0; n := 20; dev_A := series(RootOf(A-x*(A^2+2*A+2)^2, A), x = 0, n+1): seq(coeff(series(subs(A = dev_A, (1+A)*(A^2+3*A+4)*A/((3*A^2+2*A-2)^2*(A+2))), x, n+1), x, k), k = 0 .. n);

Block[{nn = 19, A, x}, A[] = 0; Do[A[x] = x*(2 + 2*A[x] + A[x]^2)^2 + O[x]^nn, nn]; CoefficientList[(1 + A[x])*(A[x]^2 + 3*A[x] + 4)* A[x]/((3*A[x]^2 + 2*A[x] - 2)^2*(A[x] + 2)), x]] (* Michael De Vlieger, Sep 03 2019 *)

G.f.: (1+A)*(A^2+3*A+4)*A/((3*A^2+2*A-2)^2*(A+2)) where A=x*(2+2*A+A^2)^2.
From Vaclav Kotesovec, Jun 25 2019: (Start)
Recurrence: 81*(n-1)*(3*n - 2)*(3*n - 1)*(20802264*n^8 - 513044308*n^7 + 5457802931*n^6 - 32703730375*n^5 + 120697828661*n^4 - 280851750277*n^3 + 402186188144*n^2 - 323841737040*n + 112137480000)*a(n) = 6*(96792934392*n^11 - 2657166947316*n^10 + 32322659739783*n^9 - 229681592172541*n^8 + 1057798736706708*n^7 - 3309904792738002*n^6 + 7166955104700747*n^5 - 10716261762345309*n^4 + 10816142222455650*n^3 - 6994792735444832*n^2 + 2594496776694720*n - 413761340160000)*a(n-1) - 32*(136213224672*n^11 - 3864805132664*n^10 + 48853431813424*n^9 - 362854015235883*n^8 + 1757540351761182*n^7 - 5820283983972594*n^6 + 13419220917200106*n^5 - 21479458450012897*n^4 + 23298284090559356*n^3 - 16214747993479962*n^2 + 6458737193497260*n - 1099216619550000)*a(n-2) - 768*(29622423936*n^11 - 845570009984*n^10 + 10735773789272*n^9 - 79940670306164*n^8 + 387373872945691*n^7 - 1280558339496068*n^6 + 2940763323423808*n^5 - 4679130395980206*n^4 + 5037190265229413*n^3 - 3476169558457578*n^2 + 1372907413337880*n - 231844115160000)*a(n-3) - 24576*(2*n - 7)*(582463392*n^10 - 14885297224*n^9 + 166341178864*n^8 - 1068833075597*n^7 + 4366030094616*n^6 - 11823901892456*n^5 + 21447449277486*n^4 - 25646549248003*n^3 + 19256170722842*n^2 - 8132937809520*n + 1445811660000)*a(n-4) - 131072*(n-5)*(2*n - 9)*(2*n - 7)*(20802264*n^8 - 346626196*n^7 + 2448956167*n^6 - 9565916473*n^5 + 22545828451*n^4 - 32733304759*n^3 + 28456182418*n^2 - 13430023272*n + 2589840000)*a(n-5).
a(n) ~ (7 + sqrt(7)) * 2^(4*n - 5) * (17 + 7*sqrt(7))^n / 3^(3*n + 1).
(End)

A385251 a(n) = Sum_{k=0..n-1} binomial(4k-3,k) binomial(4n-4k,n-k-1).

Original entry on oeis.org

0, 1, 9, 84, 790, 7452, 70401, 665692, 6298236, 59612556, 564393460, 5344664400, 50621130078, 479513718116, 4542730477758, 43039907282664, 407809863233592, 3864303038901996, 36619104142640460, 347027703183853552, 3288802989845088504, 31169274939274755312

Offset: 0

Seiichi Manyama, Jul 28 2025

nonn

Cf. A078995, A308523, A386565, A386611, A386612.

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

G.f.: (g-1)/(g * (4-3*g)^2) where g=1+x*g^4.
G.f.: g * (1-g)^2/(1-4*g)^2 where g*(1-g)^3 = x.
a(n) = Sum_{k=0..n-1} binomial(4*k-3+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n-2,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k-2,k).

cp's OEIS Frontend

A036829 a(n) = Sum_{k=0..n-1} C(3*k,k)*C(3*n-3*k-2,n-k-1).

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A386612 a(n) = Sum_{k=0..n-1} binomial(4*k+1,k) * binomial(4*n-4*k,n-k-1).

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A386368 a(n) = Sum_{k=0..n-1} binomial(6*k,k) * binomial(6*n-6*k-2,n-k-1).

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A386565 a(n) = Sum_{k=0..n-1} binomial(4*k-1,k) * binomial(4*n-4*k,n-k-1).

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A386367 a(n) = Sum_{k=0..n-1} binomial(5*k,k) * binomial(5*n-5*k-2,n-k-1).

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PARI

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A386611 a(n) = Sum_{k=0..n-1} binomial(4*k,k) * binomial(4*n-4*k,n-k-1).

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A308524 Number of essentially 3-connected rooted toroidal maps with n edges.

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Maple

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A308526 Number of essentially 3-connected rooted toroidal maps with n vertices.

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Maple

Mathematica

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A385251 a(n) = Sum_{k=0..n-1} binomial(4*k-3,k) * binomial(4*n-4*k,n-k-1).

A036829 a(n) = Sum_{k=0..n-1} C(3k,k)C(3n-3k-2,n-k-1).

A386612 a(n) = Sum_{k=0..n-1} binomial(4k+1,k) binomial(4n-4k,n-k-1).

A386368 a(n) = Sum_{k=0..n-1} binomial(6k,k) binomial(6n-6k-2,n-k-1).

A386565 a(n) = Sum_{k=0..n-1} binomial(4k-1,k) binomial(4n-4k,n-k-1).

A386367 a(n) = Sum_{k=0..n-1} binomial(5k,k) binomial(5n-5k-2,n-k-1).

A386611 a(n) = Sum_{k=0..n-1} binomial(4k,k) binomial(4n-4k,n-k-1).

A385251 a(n) = Sum_{k=0..n-1} binomial(4k-3,k) binomial(4n-4k,n-k-1).