A034261 -id:A034261 - cp's OEIS frontend

A034263 a(n) = binomial(n+4,4)(4n+5)/5.

1, 9, 39, 119, 294, 630, 1218, 2178, 3663, 5863, 9009, 13377, 19292, 27132, 37332, 50388, 66861, 87381, 112651, 143451, 180642, 225170, 278070, 340470, 413595, 498771, 597429, 711109, 841464, 990264, 1159400, 1350888, 1566873, 1809633, 2081583, 2385279

Offset: 0

Clark Kimberling, Barry E. Williams, Dec 13 1999

Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
5-dimensional form of hexagonal-based pyramid numbers. - Ben Creech (mathroxmysox(AT)yahoo.com), Nov 17 2005
Convolution of triangular numbers (A000217) and hexagonal numbers (A000384). - Bruno Berselli, Jun 27 2013

			By the third comment: A000217(1..6) and A000384(1..6) give the term a(5) = 1*21+5*15+12*10+22*6+35*3+51*1 = 630. - _Bruno Berselli_, Jun 27 2013

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 167-169, Table 10.5/II/4).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
Index to sequences related to pyramidal numbers.

Partial sums of A002417.
Cf. A000217, A000384, A000292, A034261, A059599, A093561.
Cf. similar sequences listed in A254142.

List([0..35], n-> (4*n+5)*Binomial(n+4,4)/5); # G. C. Greubel, Aug 28 2019

[(4*n+5)*Binomial(n+4,4)/5: n in [0..35]]; // G. C. Greubel, Aug 28 2019

a:=n->(n+1)*(n+2)*(n+3)*(n+4)*(4*n+5)/120: seq(a(n),n=0..35); # Emeric Deutsch, Nov 18 2005

Table[Binomial[n+4, 4]*(4*n+5)/5, {n,0,35}] (* Vladimir Joseph Stephan Orlovsky, Jan 26 2012 *)
a[n_] := (1+n)(2+n)(3+n)(4+n)(4n+5)/120; Array[a, 36, 0] (* or *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 9, 39, 119, 294, 630}, 36] (* or *)
CoefficientList[ Series[(1+3*x)/(1-x)^6, {x, 0, 35}], x] (* Robert G. Wilson v, Feb 26 2015 *)
Table[Sum[-x^2 + y^2 + z^2, {x, 0, g}, {y, x, g}, {z, y, g}], {g, 1, 30}]/4 (* Horst H. Manninger, Jun 19 2025 *)

a(n)=(n+1)*(n+2)*(n+3)*(n+4)*(4*n+5)/120 \\ Charles R Greathouse IV, Sep 24 2015, corrected by Altug Alkan, Aug 15 2017

[(4*n+5)*binomial(n+4,4)/5 for n in (0..35)] # G. C. Greubel, Aug 28 2019

a(n) = A093561(n+5, 5).
a(n) = A034261(n+1, 3).
G.f.: (1+3*x)/(1-x)^6.
a(n) = (n+1)*(n+2)*(n+3)*(n+4)*(4*n+5)/120. - Emeric Deutsch and Ben Creech (mathroxmysox(AT)yahoo.com), Nov 17 2005, corrected by Eric Rowland, Aug 15 2017
a(-n-4) = -A059599(n). - Bruno Berselli, Aug 23 2011
a(n) = Sum_{i=1..n+1} i*A000292(i). - Bruno Berselli, Jan 23 2015
Sum_{n>=0} 1/a(n) = 28300/231 - 1280*Pi/77 - 7680*log(2)/77. - Amiram Eldar, Feb 15 2022

Corrected and extended by N. J. A. Sloane, Apr 21 2000

A089574 Column 4 of an array closely related to A083480. (Both arrays have shape sequence A083479).

Original entry on oeis.org

5, 32, 113, 299, 664, 1309, 2366, 4002, 6423, 9878, 14663, 21125, 29666, 40747, 54892, 72692, 94809, 121980, 155021, 194831, 242396, 298793, 365194, 442870, 533195, 637650, 757827, 895433, 1052294, 1230359, 1431704, 1658536, 1913197

Offset: 1

Alford Arnold, Dec 29 2003; extended May 04 2005

The diagonals are finite and sum to A047970.
Values appear to be a transformation of A006468 (rooted planar maps). Also known as well-labeled trees (cf. A000168).
First differences of the conjectured polynomial formula for A006468. [From R. J. Mathar, Jun 26 2010]

			The array begins
1
2
4
7 1
11 5
16 14 2
22 30 12
29 55 39 5
37 91 95 32 1

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A105552, A006468.
Cf. A006011, A034261.
Cf. A000124 (column 1), A000330 (column 2), A086602 (column 3), A107600 (column 5), A107601 (column 6), A109125 (column 7), A109126 (column 8), A109820 (column 9), A108538 (column 10), A109821 (column 11), A110553 (column 12), A110624 (column 13).

Row sums are powers of 2.
a(n) = A000330(n) + A006011(n+1) + A034263(n-1).
a(n)= +6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6). G.f.: x*(5+2*x-4*x^2+x^3)/(x-1)^6. a(n) = n*(n+1)*(4*n^3+51*n^2+159*n+86)/120. [From R. J. Mathar, Jun 26 2010]

Extended beyond a(8) by R. J. Mathar, Jun 26 2010

A002055 Number of diagonal dissections of a convex n-gon into n-4 regions.

Original entry on oeis.org

1, 9, 56, 300, 1485, 7007, 32032, 143208, 629850, 2735810, 11767536, 50220040, 212952285, 898198875, 3771484800, 15775723920, 65770848990, 273420862110, 1133802618000, 4691140763400, 19371432850770, 79850555673174

Offset: 5

N. J. A. Sloane

Number of standard tableaux of shape (n-4,n-4,1,1) (see Stanley reference). - Emeric Deutsch, May 20 2004
Number of increasing tableaux of shape (n-2,n-2) with largest entry 2n-6. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. - Oliver Pechenik, May 02 2014
a(n) = number of noncrossing partitions of 2n-6 into n-4 blocks, each of size at least 2. - Oliver Pechenik, May 02 2014

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 5..100
D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.
A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
O. Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, arXiv:1209.1355 [math.CO], 2012-2014.
O. Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, J. Combin. Theory A, 125 (2014), 357-378.
R. C. Read, On general dissections of a polygon, Preprint (1974)
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388, Table 1.
R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.

a(n) = f(n,n+1) where f is given in A034261.

Table[(Binomial[n-3,2]Binomial[2n-6,n-5])/(n-4),{n,5,30}] (* Harvey P. Dale, Nov 06 2011 *)

a(n) = (binomial(n - 3, 2) * binomial(2*n - 6, n - 5))/(n - 4);
for(n=5, 30, print1(a(n),", ")) \\ Indranil Ghosh, Apr 11 2017

a(n) = binomial(n-3, 2)*binomial(2*n-6, n-5)/(n-4).
With offset 0, this has a(n)=(n+2)*C(2n+4,n)/2 and e.g.f. dif(dif(x*dif(exp(2x)*Bessel_I(2,2x),x),x),x)/2. - Paul Barry, Aug 25 2007
G.f.: 16*x^5*(x+sqrt(1-4x))/((1-4x)^(3/2) *(1+sqrt(1-4x))^4 ). - R. J. Mathar, Nov 17 2011
D-finite with recurrence: (n-1)*a(n) +(23-11n)*a(n-1) +10*(4n-13)*a(n-2) +10*(23-5n)*a(n-3) +4*(2n-13)*a(n-4)=0. - R. J. Mathar, Nov 17 2011
a(n) ~ 4^n*sqrt(n)/(128*sqrt(Pi)). - Ilya Gutkovskiy, Apr 11 2017

A034265 a(n) = binomial(n+6,6)(6n+7)/7.

Original entry on oeis.org

1, 13, 76, 300, 930, 2442, 5676, 12012, 23595, 43615, 76648, 129064, 209508, 329460, 503880, 751944, 1097877, 1571889, 2211220, 3061300, 4177030, 5624190, 7480980, 9839700, 12808575, 16513731, 21101328, 26739856, 33622600, 41970280

Offset: 0

Clark Kimberling

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

a(n)=f(n, 5) where f is given in A034261.
Partial sums of A027810.
Cf. A093563 ((6, 1) Pascal, column m=7).
Cf. similar sequences listed in A254142.

List([0..30], n-> (6*n+7)*Binomial(n+6,6)/7); # G. C. Greubel, Aug 28 2019

[(6*n+7)*Binomial(n+6,6)/7: n in [0..40]]; // Vincenzo Librandi, Jul 30 2014

seq((6*n+7)*binomial(n+6,6)/7, n=0..30); # G. C. Greubel, Aug 28 2019

Accumulate[Table[(n+1)Binomial[n+5,5],{n,0,30}]] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1}, {1,13,76,300,930,2442,5676, 12012}, 30] (* Harvey P. Dale, Jul 29 2014 *)
CoefficientList[Series[(1+5x)/(1-x)^8, {x,0,40}], x] (* Vincenzo Librandi, Jul 30 2014 *)

a(n)=(6*n/7+1)*binomial(n+6,6) \\ Charles R Greathouse IV, Oct 07 2015

[(6*n+7)*binomial(n+6,6)/7 for n in (0..30)] # G. C. Greubel, Aug 28 2019

G.f.: (1+5*x)/(1-x)^8.

a(0)=1, a(1)=13, a(2)=76, a(3)=300, a(4)=930, a(5)=2442, a(6)=5676, a(7)=12012, a(n) = 8*a(n-1) -28*a(n-2) +56*a(n-3) -70*a(n-4) +56*a(n-5) -28*a(n-6) +8*a(n-7) -a(n-8). - Harvey P. Dale, Jul 29 2014

Corrected and extended by N. J. A. Sloane, Apr 21 2000

A034266 Partial sums of A027818.

Original entry on oeis.org

0, 1, 15, 99, 435, 1485, 4257, 10725, 24453, 51480, 101530, 189618, 338130, 579462, 959310, 1540710, 2408934, 3677355, 5494401, 8051725, 11593725, 16428555, 22940775, 31605795, 43006275, 57850650, 76993956, 101461140, 132473044, 171475260, 220170060, 280551612

Offset: 0

Clark Kimberling

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, pp. 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000

a(n)=f(n, 6) where f is given in A034261.
Cf. A027818, A053367, A034266.
Cf. A093564 ((7, 1) Pascal, column m=8).
Cf. similar sequences listed in A254142.

List([0..35], n-> (7*n+1)*Binomial(n+6,7)/8); # G. C. Greubel, Aug 29 2019

[0] cat [(7*n+8)*Binomial(n+7, 7)/8: n in [0..30]]; // Vincenzo Librandi, Mar 20 2015

f:=n->(7*n+8)*binomial(n+7, 7)/8; [seq(f(n),n=-1..40)]; # N. J. A. Sloane, Mar 25 2015

CoefficientList[Series[x(1+6x)/(1-x)^9, {x, 0, 30}], x] (* Vincenzo Librandi, Mar 20 2015 *)
Table[(7*n+1)*Binomial[n+6,7]/8, {n,0,35}] (* G. C. Greubel, Aug 29 2019 *)

lista(nn) = for (n=0, nn, print1((7*n+1)*binomial(n+6,7)/8, ", ")); \\ Michel Marcus, Mar 20 2015

[(7*n+1)*binomial(n+6,7)/8 for n in (0..35)] # G. C. Greubel, Aug 29 2019

a(n) = (7*n+1)*binomial(n+6, 7)/8.
G.f.: x*(1+6*x)/(1-x)^9.
E.g.f.: x*(8! +262080*x +383040*x^2 +210000*x^3 +52080*x^4 +6216*x^5 + 344*x^6 +7*x^7)*exp(x)/8!

Better description from Barry E. Williams, Jan 25 2000
Corrected and extended by N. J. A. Sloane, Apr 21 2000
More terms from Michel Marcus, Mar 20 2015

cp's OEIS Frontend

A034263 a(n) = binomial(n+4,4)*(4*n+5)/5.

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A089574 Column 4 of an array closely related to A083480. (Both arrays have shape sequence A083479).

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A002055 Number of diagonal dissections of a convex n-gon into n-4 regions.

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A034265 a(n) = binomial(n+6,6)*(6*n+7)/7.

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GAP

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A034266 Partial sums of A027818.

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GAP

Magma

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Extensions

A034263 a(n) = binomial(n+4,4)(4n+5)/5.

A034265 a(n) = binomial(n+6,6)(6n+7)/7.