A051843 -id:A051843 - cp's OEIS frontend

A005585 5-dimensional pyramidal numbers: a(n) = n(n+1)(n+2)(n+3)(2n+3)/5!.

1, 7, 27, 77, 182, 378, 714, 1254, 2079, 3289, 5005, 7371, 10556, 14756, 20196, 27132, 35853, 46683, 59983, 76153, 95634, 118910, 146510, 179010, 217035, 261261, 312417, 371287, 438712, 515592, 602888, 701624, 812889, 937839, 1077699, 1233765, 1407406

Offset: 1

N. J. A. Sloane

Convolution of triangular numbers (A000217) and squares (A000290) (n>=1). - Graeme McRae, Jun 07 2006
p^k divides a(p^k-3), a(p^k-2), a(p^k-1) and a(p^k) for prime p > 5 and integer k > 0. p^k divides a((p^k-3)/2) for prime p > 5 and integer k > 0. - Alexander Adamchuk, May 08 2007
If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-5) is the number of 6-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
5-dimensional square numbers, fourth partial sums of binomial transform of [1,2,0,0,0,...]. a(n) = Sum_{i=0..n} binomial(n+4, i+4)*b(i), where b(i)=[1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
Antidiagonal sums of the convolution array A213550. - Clark Kimberling, Jun 17 2012
Binomial transform of (1, 6, 14, 16, 9, 2, 0, 0, 0, ...). - Gary W. Adamson, Jul 28 2015
2*a(n) is number of ways to place 4 queens on an (n+3) X (n+3) chessboard so that they diagonally attack each other exactly 6 times. The maximal possible attack number, p=binomial(k,2)=6 for k=4 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form a corresponding complete graph. - Antal Pinter, Dec 27 2015
While adjusting for offsets, add A000389 to find the next in series A000389, A005585, A051836, A034263, A027800, A051843, A051877, A051878, A051879, A051880, A056118, A271567. (See Bruno Berselli's comments in A271567.) - Bruce J. Nicholson, Jun 21 2018
Coefficients in the terminating series identity 1 - 7*n/(n + 6) + 27*n*(n - 1)/((n + 6)*(n + 7)) - 77*n*(n - 1)*(n - 2)/((n + 6)*(n + 7)*(n + 8)) + ... = 0 for n = 1,2,3,.... Cf. A002415 and A040977. - Peter Bala, Feb 18 2019

			G.f. = x + 7*x^2 + 27*x^3 + 77*x^4 + 182*x^5 + 378*x^6 + 714*x^7 + 1254*x^8 + ... - _Michael Somos_, Jun 24 2018

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 797.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000 (first 121 terms from Alexander Adamchuk)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
R. K. Guy, Letter to N. J. A. Sloane, Feb 1988
Milan Janjic, Two Enumerative Functions
C. H. Karlson and N. J. A. Sloane, Correspondence, 1974
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
R. P. Stanley and F. Zanello, The Catalan case of Armstrong's conjecture on core partitions, arXiv preprint arXiv:1312.4352 [math.CO], 2013.
Index entries for sequences related to Chebyshev polynomials.
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

a(n) = ((-1)^(n+1))*A053120(2*n+3, 5)/16, (1/16 of sixth unsigned column of Chebyshev T-triangle, zeros omitted).
Partial sums of A002415.
Cf. A006542, A040977, A047819, A111125 (third column).
Cf. a(n) = ((-1)^(n+1))*A084960(n+1, 2)/16 (compare with the first line). - Wolfdieter Lang, Aug 04 2014
Cf. A000389, A051836, A034263, A027800, A051843, A051877, A051878, A051879.

I:=[1, 7, 27, 77, 182, 378]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..40]]; // Vincenzo Librandi, Jun 09 2013

[seq(binomial(n+2,6)-binomial(n,6), n=4..45)]; # Zerinvary Lajos, Jul 21 2006
A005585:=(1+z)/(z-1)**6; # Simon Plouffe in his 1992 dissertation

With[{c=5!},Table[n(n+1)(n+2)(n+3)(2n+3)/c,{n,40}]] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{1,7,27,77,182,378},40] (* Harvey P. Dale, Oct 04 2011 *)
CoefficientList[Series[(1 + x) / (1 - x)^6, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)

a(n)=binomial(n+3,4)*(2*n+3)/5 \\ Charles R Greathouse IV, Jul 28 2015

G.f.: x*(1+x)/(1-x)^6.
a(n) = 2*C(n+4, 5) - C(n+3, 4). - Paul Barry, Mar 04 2003
a(n) = C(n+3, 5) + C(n+4, 5). - Paul Barry, Mar 17 2003
a(n) = C(n+2, 6) - C(n, 6), n >= 4. - Zerinvary Lajos, Jul 21 2006
a(n) = Sum_{k=1..n} T(k)*T(k+1)/3, where T(n) = n(n+1)/2 is a triangular number. - Alexander Adamchuk, May 08 2007
a(n-1) = (1/4)*Sum_{1 <= x_1, x_2 <= n} |x_1*x_2*det V(x_1,x_2)| = (1/4)*Sum_{1 <= i,j <= n} i*j*|i-j|, where V(x_1,x_2) is the Vandermonde matrix of order 2. First differences of A040977. - Peter Bala, Sep 21 2007
a(n) = C(n+4,4) + 2*C(n+4,5). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
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), a(1)=1, a(2)=7, a(3)=27, a(4)=77, a(5)=182, a(6)=378. - Harvey P. Dale, Oct 04 2011
a(n) = (1/6)*Sum_{i=1..n+1} (i*Sum_{k=1..i} (i-1)*k). - Wesley Ivan Hurt, Nov 19 2014
E.g.f.: x*(2*x^4 + 35*x^3 + 180*x^2 + 300*x + 120)*exp(x)/120. - Robert Israel, Nov 19 2014
a(n) = A000389(n+3) + A000389(n+4). - Bruce J. Nicholson, Jun 21 2018
a(n) = -a(-3-n) for all n in Z. - Michael Somos, Jun 24 2018
From Amiram Eldar, Jun 28 2020: (Start)
Sum_{n>=1} 1/a(n) = 40*(16*log(2) - 11)/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 20*(8*Pi - 25)/3. (End)
a(n) = A004302(n+1) - A207361(n+1). - J. M. Bergot, May 20 2022
a(n) = Sum_{i=0..n+1} Sum_{j=i..n+1} i*j*(j-i)/2. - Darío Clavijo, Oct 11 2023
a(n) = (A000538(n+1) - A000330(n+1))/12. - Yasser Arath Chavez Reyes, Feb 21 2024

A034827 a(n) = 2*binomial(n,4).

Original entry on oeis.org

0, 0, 0, 0, 2, 10, 30, 70, 140, 252, 420, 660, 990, 1430, 2002, 2730, 3640, 4760, 6120, 7752, 9690, 11970, 14630, 17710, 21252, 25300, 29900, 35100, 40950, 47502, 54810, 62930, 71920, 81840, 92752, 104720, 117810, 132090, 147630, 164502, 182780

Offset: 0

N. J. A. Sloane

Also number of ways to insert two pairs of parentheses into a string of n-4 letters (allowing empty pairs of parentheses). E.g., there are 30 ways for 2 letters. Cf. A002415.
2,10,30,70, ... gives orchard crossing number of complete graph K_n. - Ralf Stephan, Mar 28 2003
If Y is a 2-subset of an n-set X then, for n>=4, a(n-1) is the number of 4-subsets and 5-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007
Middle column of table on p. 6 of Feder and Garber. - Jonathan Vos Post, Apr 23 2009
Number of pairs of non-intersecting lines when each of n points around a circle is joined to every other point by straight lines. A pair of lines is considered non-intersecting if the lines do not intersect in either the interior or the boundary of a circle. - Melvin Peralta, Feb 05 2016
From a(2), convolution of the oblong numbers (A002378) with the nonnegative numbers (A001477). - Bruno Berselli, Oct 24 2016
Also the number of 3-cycles in the n-triangular honeycomb bishop graph. - Eric W. Weisstein, Aug 10 2017

Charles Jordan, Calculus of Finite Differences, Chelsea, 1965, p. 449.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Bruno Berselli, Table of n, a(n) for n = 0..1000
M. Aganagic, A. Klemm and C. Vafa, Disk Instantons, Mirror Symmetry and the Duality Web, arXiv:hep-th/0105045, 2001.
Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
Elie Feder and David Garber, The Orchard crossing number of an abstract graph, arXiv:math/0303317 [math.CO], 2003-2009.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), pp. 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Eric Weisstein's World of Mathematics, Graph Cycle.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

A diagonal of A088617.
Cf. A033487, A050534, A060008.
Partial sums of A007290.
Cf. A001477, A002378.
Cf. A051843 (4-cycles in the triangular honeycomb bishop graph), A290775 (5-cycles), A290779 (6-cycles).

[2*Binomial(n,4): n in [0..40]]; // Vincenzo Librandi, Oct 20 2013

[seq(binomial(n,4)*2,n=0..40)]; # Zerinvary Lajos, Jul 18 2006

CoefficientList[Series[2 x^4/(1 - x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Oct 20 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 0, 0, 2}, 50] (* Harvey P. Dale, Jun 09 2016 *)
Table[2 Binomial[n, 4], {n, 0, 40}] (* Bruno Berselli, Oct 24 2016 *)
2 Binomial[Range[0, 20], 4] (* Eric W. Weisstein, Aug 10 2017 *)

a(n)=2*binomial(n,4) \\ Charles R Greathouse IV, Jun 23 2015

a(n) = A096338(2*n-6) = 2*A000332(n), n>2. - R. J. Mathar, Nov 08 2010
G.f.: 2*x^4/(1-x)^5. - Colin Barker, Feb 29 2012
a(n) = Sum_{k=1..n-3} ( Sum_{i=1..k} i*(2*k-n+4) ). - Wesley Ivan Hurt, Sep 26 2013
E.g.f.: x^4*exp(x)/12. - G. C. Greubel, Feb 23 2017
From Amiram Eldar, Jul 19 2022: (Start)
Sum_{n>=4} 1/a(n) = 2/3.
Sum_{n>=4} (-1)^n/a(n) = 16*log(2) - 32/3. (End)

A093563 (6,1)-Pascal triangle.

Original entry on oeis.org

1, 6, 1, 6, 7, 1, 6, 13, 8, 1, 6, 19, 21, 9, 1, 6, 25, 40, 30, 10, 1, 6, 31, 65, 70, 40, 11, 1, 6, 37, 96, 135, 110, 51, 12, 1, 6, 43, 133, 231, 245, 161, 63, 13, 1, 6, 49, 176, 364, 476, 406, 224, 76, 14, 1, 6, 55, 225, 540, 840, 882, 630, 300, 90, 15, 1, 6, 61, 280, 765, 1380

Offset: 0

Wolfdieter Lang, Apr 22 2004

The array F(6;n,m) gives in the columns m >= 1 the figurate numbers based on A016921, including the octagonal numbers A000567, (see the W. Lang link).
This is the sixth member, d=6, in the family of triangles of figurate numbers, called (d,1) Pascal triangles: A007318 (Pascal), A029653, A093560-2, for d=1..5.
This is an example of a Riordan triangle (see A093560 for a comment and A053121 for a comment and the 1991 Shapiro et al. reference on the Riordan group). Therefore the o.g.f. for the row polynomials p(n,x):=Sum_{m=0..n} a(n,m)*x^m is G(z,x)=(1+5*z)/(1-(1+x)*z).
The SW-NE diagonals give A022096(n-1) = Sum_{k=0..ceiling((n-1)/2)} a(n-1-k,k), n >= 1, with n=0 value 5. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs.
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013

			Triangle begins
  1;
  6,  1;
  6,  7,  1;
  6, 13,  8,  1;
  6, 19, 21,  9,  1;
  6, 25, 40, 30, 10,  1;
  ...

Kurt Hawlitschek, Johann Faulhaber 1580-1635, Veroeffentlichung der Stadtbibliothek Ulm, Band 18, Ulm, Germany, 1995, Ch. 2.1.4. Figurierte Zahlen.
Ivo Schneider: Johannes Faulhaber 1580-1635, Birkhäuser, Basel, Boston, Berlin, 1993, ch.5, pp. 109-122.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Wolfdieter Lang, First 10 rows and array of figurate numbers

Row sums: A005009(n-1), n>=1, 1 for n=0, alternating row sums are 1 for n=0, 5 for n=2 and 0 else.
Cf. A007318, A093564 (d=7), A228196, A228576.
The column sequences give for m=1..9: A016921, A000567 (octagonal), A002414, A002419, A051843, A027810, A034265, A054487, A055848.

a093563 n k = a093563_tabl !! n !! k
a093563_row n = a093563_tabl !! n
a093563_tabl = [1] : iterate
               (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [6, 1]
-- Reinhard Zumkeller, Aug 31 2014

lim = 11; s = Series[(1 + 5*x)/(1 - x)^(m + 1), {x, 0, lim}]; t = Table[ CoefficientList[s, x], {m, 0, lim}]; Flatten[ Table[t[[j - k + 1, k]], {j, lim + 1}, {k, j, 1, -1}]] (* Jean-François Alcover, Sep 16 2011, after g.f. *)

from math import comb, isqrt
def A093563(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),a:=n-comb(r+1,2))*(r+5*(r-a))//r if n else 1 # Chai Wah Wu, Nov 12 2024

a(n, m)=F(6;n-m, m) for 0<= m <= n, otherwise 0, with F(6;0, 0)=1, F(6;n, 0)=6 if n>=1 and F(6;n, m):= (6*n+m)*binomial(n+m-1, m-1)/m if m>=1.
Recursion: a(n, m)=0 if m>n, a(0, 0)= 1; a(n, 0)=6 if n>=1; a(n, m)= a(n-1, m) + a(n-1, m-1).
G.f. column m (without leading zeros): (1+5*x)/(1-x)^(m+1), m>=0.
T(n, k) = C(n, k) + 5*C(n-1, k). - Philippe Deléham, Aug 28 2005
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(6 + 13*x + 8*x^2/2! + x^3/3!) = 6 + 19*x + 40*x^2/2! + 70*x^3/3! + 110*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

A027810 a(n) = (n+1)*binomial(n+5, 5).

Original entry on oeis.org

1, 12, 63, 224, 630, 1512, 3234, 6336, 11583, 20020, 33033, 52416, 80444, 119952, 174420, 248064, 345933, 474012, 639331, 850080, 1115730, 1447160, 1856790, 2358720, 2968875, 3705156, 4587597, 5638528, 6882744, 8347680, 10063592, 12063744, 14384601, 17066028

Offset: 0

Thi Ngoc Dinh (via R. K. Guy)

Number of 11-subsequences of [ 1, n ] with just 5 contiguous pairs.

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Herbert John Ryser, Combinatorial Mathematics, Carus Mathematical Monographs No. 14, John Wiley and Sons, 1963, pp. 1-8.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Partial sums of A051843.
Cf. A093563 ((6, 1) Pascal, column m=6).
Cf. A007318, A000142, A245334.

a027810 n = (n + 1) * a007318' (n + 5) 5
-- Reinhard Zumkeller, Aug 31 2014

[(n+1)*Binomial(n+5,5): n in [0..40]]; // Vincenzo Librandi, Jul 30 2014

[n*(n-1)*(n-2)*(n-3)*(n-4)^2/120: n in [5..40]]; // Vincenzo Librandi, Jul 30 2014

[seq(n*(n-1)*(n-2)*(n-3)*(n-4)^2/5!,n=5..33)]; # Zerinvary Lajos, Oct 19 2006

Table[(n+1)Binomial[n+5,5],{n,0,30}] (* Harvey P. Dale, Jul 29 2014 *)
CoefficientList[Series[(1 + 5 x)/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 30 2014 *)

a(n)=n*(n^5+16*n^4+100*n^3+310*n^2+499*n+394)/120+1 \\ Charles R Greathouse IV, Sep 28 2015

G.f.: (1+5*x)/(1-x)^7.
a(n) = A245334(n+5, 5)/A000142(5). - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 5*Pi^2/6 - 1025/144.
Sum_{n>=0} (-1)^n/a(n) = 5*Pi^2/12 - 160*log(2)/3 + 4865/144. (End)

Two redundant formulas deleted by N. J. A. Sloane, Jul 30 2014

A290775 Number of 5-cycles in the n-triangular honeycomb bishop graph.

Original entry on oeis.org

0, 0, 2, 24, 138, 532, 1596, 4032, 8988, 18216, 34254, 60632, 102102, 164892, 256984, 388416, 571608, 821712, 1156986, 1599192, 2174018, 2911524, 3846612, 5019520, 6476340, 8269560, 10458630, 13110552, 16300494, 20112428, 24639792, 29986176, 36266032, 43605408, 52142706

Offset: 1

Eric W. Weisstein, Aug 10 2017

Eric Weisstein's World of Mathematics, Graph Cycle
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A034827 (3-cycles in the triangular honeycomb bishop graph), A051843 (4-cycles), A290779 (6-cycles).

Table[2/5 Binomial[n + 1, 4] (8 - 7 n + 2 n^2), {n, 20}]
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 0, 2, 24, 138, 532, 1596}, 20]
CoefficientList[Series[-((2 (x^2 + 5 x^3 + 6 x^4))/(-1 + x)^7), {x, 0, 20}], x]

a(n)=n*(2*n^5 - 11*n^4 + 20*n^3 - 5*n^2 - 22*n + 16)/60 \\ Charles R Greathouse IV, Aug 10 2017

a(n) = 2/5 * binomial(n + 1, 4)*(8 - 7*n + 2*n^2).
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).
G.f.: -((2 x (x^2 + 5 x^3 + 6 x^4))/(-1 + x)^7).

A290779 Number of 6-cycles in the n-triangular honeycomb bishop graph.

Original entry on oeis.org

0, 0, 1, 57, 486, 2360, 8394, 24354, 61104, 137412, 283635, 546403, 994422, 1725516, 2875028, 4625700, 7219152, 10969080, 16276293, 23645709, 33705430, 47228016, 65154078, 88618310, 118978080, 157844700, 207117495, 269020791, 346143942, 441484516, 558494760

Offset: 1

Eric W. Weisstein, Aug 10 2017

Eric Weisstein's World of Mathematics, Graph Cycle
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A034827 (3-cycles), A051843 (4-cycles), A290775 (5-cycles).

Table[Binomial[n + 1, 4] (-62 + 11 n - 109 n^2 + 40 n^3)/70, {n, 20}]
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 0, 1, 57, 486, 2360, 8394, 24354}, 40]
CoefficientList[Series[(x^2 + 49 x^3 + 58 x^4 + 12 x^5)/(-1 + x)^8, {x, 0, 20}], x]

a(n)=n*(40*n^6 - 189*n^5 + 189*n^4 + 105*n^3 - 105*n^2 + 84*n - 124)/1680 \\ Charles R Greathouse IV, Aug 10 2017

a(n) = binomial(n + 1, 4)*(-62 + 11*n - 109*n^2 + 40*n^3)/70.
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).
G.f.: (x (x^2 + 49 x^3 + 58 x^4 + 12 x^5))/(-1 + x)^8.

A136526 Coefficients polynomials B(x, n) = ((1 + a + b)x - c)B(x, n-1) - abB(x, n-2) with a = 3, b = 2, and c = 0.

Original entry on oeis.org

1, 0, 1, -6, 0, 6, 0, -42, 0, 36, 36, 0, -288, 0, 216, 0, 468, 0, -1944, 0, 1296, -216, 0, 4536, 0, -12960, 0, 7776, 0, -4104, 0, 38880, 0, -85536, 0, 46656, 1296, 0, -51840, 0, 311040, 0, -559872, 0, 279936, 0, 32400, 0, -544320, 0, 2379456, 0, -3639168, 0, 1679616

Offset: 0

Roger L. Bagula, Mar 23 2008

			Triangle begins as:
     1;
     0,     1;
    -6,     0,      6;
     0,   -42,      0,    36;
    36,     0,   -288,     0,    216;
     0,   468,      0, -1944,      0,   1296;
  -216,     0,   4536,     0, -12960,      0,    7776;
     0, -4104,      0, 38880,      0, -85536,       0, 46656;
  1296,     0, -51840,     0, 311040,      0, -559872,     0, 279936;

Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, page 93

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000567, A002414, A002419, A016921, A027810, A030192, A034265, A051843, A054487, A055848, A081106, A136531.

f:= func< n,k | k eq 0 select (-1)^Floor(n/2) else (-1)^Floor((n-k)/2)*6^Floor((k-1)/2)*(1/k)*(6*Floor((n-k)/2) +k)*Binomial(Floor((n-k)/2) +k-1, k-1) >;
A136526:= func< n,k | ((n+k+1) mod 2)*6^Floor(n/2)*f(n,k) >;
[A136526(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 22 2022

(* First program *)
a= (b+1)/(b-1); c=0; b=2;
B[x_, n_]:= B[x, n]= If[n<2, x^n, ((1+a+b)*x -c)*B[x, n-1] -a*b*B[x, n-2]];
Table[CoefficientList[B[x,n], x], {n,0,10}]//Flatten
(* Second program *)
B[x_, n_]:= 6^(n/2)*(ChebyshevU[n, Sqrt[3/2]*x] -(5*x/Sqrt[6])*ChebyshevU[n-1, Sqrt[3/2]*x]);
Table[CoefficientList[B[x, n], x]/6^Floor[n/2], {n,0,16}]//Flatten (* G. C. Greubel, Sep 22 2022 *)

def f(n,k):
    if (k==0): return (-1)^(n//2)
    else: return (-1)^((n-k)//2)*6^((k-1)//2)*(1/k)*(6*((n-k)//2) + k)*binomial(((n-k)//2) +k-1, k-1)
def A136526(n,k): return ((n+k+1)%2)*6^(n//2)*f(n,k)
flatten([[A136526(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 22 2022

T(n, k) = coefficients of the polynomials defined by B(x, n) = ((1 + a + b)*x - c)*B(x, n - 1) - a*b*B(x, n - 2) with B(x, 0) = 1, B(x, 1) = x, a = 3, b = 2, and c = 0.
From G. C. Greubel, Sep 22 2022: (Start)
T(n, k) = coefficients of the polynomials defined by B(x, n) = 6^(n/2)*(ChebyshevU(n, sqrt(3/2)*x) - (5*x/sqrt(6))*ChebyshevU(n-1, sqrt(3/2)*x)).
T(n, k) = (1/2)*(1+(-1)^(n+k))*6^floor(n/2)*f(n, k), where f(n, k) = (-1)^floor((n -k)/2)*6^floor((k-1)/2)*(1/k)*(6*floor((n-k)/2) + k)*binomial(floor((n-k)/2) + k -1, k-1), for k >= 1, and (-1)^floor(n/2) for k = 0.
T(n, 0) = (1/2)*(1+(-1)^n)*(-6)^floor(n/2).
T(n, 1) = (1/2)*(1-(-1)^n)*(-6)^floor((n-1)/2)*A016921(floor((n-1)/2)), n >= 1.
T(n, 2) = (1/2)*(1+(-1)^n)*(-1)^(1+Floor((n+1)/2))*6^floor((n+1)/2)*A000567(floor( (n+1)/2)), n >= 2.
T(n, 3) = (1/2)*(1-(-1)^n)*(-6)^floor((n+1)/2)*A002414(floor((n-1)/2)), n >= 3.
T(n, 4) = (3/2)*(1+(-1)^n)*(-6)^floor((n+1)/2)*A002419(floor((n-1)/2)), n >= 4.
T(n, 5) = 18*(1-(-1)^n)*(-6)^floor((n-1)/2)*A051843(floor((n-3)/2)), n >= 5.
T(n, n) = 6^(n-1) + (5/6)*[n=0].
T(n, n-2) = -6*A081106(n-2), n >= 2.
Sum_{k=0..n} T(n, k) = -6*A030192(n-3), n>= 0.
Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] - 5*[n=2].
G.f.: (1 - 5*x*y)/(1 - 6*x*y + 6*y^2). (End)

Edited by G. C. Greubel, Sep 22 2022

A125235 Triangle with the partial column sums of the octagonal numbers.

Original entry on oeis.org

1, 8, 1, 21, 9, 1, 40, 30, 10, 1, 65, 70, 40, 11, 1, 96, 135, 110, 51, 12, 1, 133, 231, 245, 161, 63, 13, 1, 176, 364, 476, 406, 224, 76, 14, 1, 225, 540, 840, 882, 630, 300, 90, 15, 1, 280, 765, 1380, 1722, 1512, 930, 390, 105, 16, 1

Offset: 1

Gary W. Adamson, Nov 24 2006

"Partial column sums" means the octagonal numbers are the 1st column, the 2nd column are the partial sums of the 1st column, the 3rd column are the partial sums of the 2nd, etc.

Row sums are 1, 9, 31, 81, 187, 405, 847 = 7*(2^n-1) - 6*n. - R. J. Mathar, Sep 06 2011

			First few rows of the triangle:
   1;
   8,   1;
  21,   9,   1;
  40,  30,  10,   1;
  65,  70,  40,  11,   1;
  96, 135, 110,  51,  12,   1;
  ...

Albert H. Beiler, Recreations in the Theory of Numbers, Dover (1966), p. 189.

Cf. A000567, A002414, A002419, A051843, A027810, A125232 - A125234.

t(n, k) = if (n <0, 0, if (k==1, n*(3*n-2), if (k > 1, t(n-1,k-1) + t(n-1,k))));
tabl(nn) = {for (n = 1, nn, for (k = 1, n, print1(t(n, k), ", ");); print(););} \\ Michel Marcus, Mar 04 2014

T(n,1) = A000567(n).
T(n,k) = T(n-1,k-1) + T(n-1,k), k>1.
T(n,2) = A002414(n-1).
T(n,3) = A002419(n-2).
T(n,4) = A051843(n-4).
T(n,5) = A027810(n-6).

More terms from Michel Marcus, Mar 04 2014

A271567 Convolution of nonzero triangular numbers (A000217) and nonzero tetradecagonal numbers (A051866).

Original entry on oeis.org

1, 17, 87, 287, 742, 1638, 3234, 5874, 9999, 16159, 25025, 37401, 54236, 76636, 105876, 143412, 190893, 250173, 323323, 412643, 520674, 650210, 804310, 986310, 1199835, 1448811, 1737477, 2070397, 2452472, 2888952, 3385448, 3947944, 4582809, 5296809, 6097119

Offset: 0

Ilya Gutkovskiy, Apr 12 2016

More generally, the ordinary generating function for the convolution of triangular numbers and k-gonal numbers is (1 + (k - 3)*x)/(1 - x)^6.

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Triangular Number
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1)

Cf. A000217, A051866.

Cf. similar sequences of the convolution of triangular numbers and k-gonal numbers: A005585 (k=4), A051836 (k=5), A034263 (k=6), A027800 (k=7), A051843 (k=8), A051877 (k=9), A051878 (k=10), A051879 (k=11), A051880 (k=12), A056118 (k=13), this sequence (k=14).

/* From definition: */ P:=func; /*, where P(n, k) is the n-th k-gonal number, */ [&+[P(n+1-i, 3)*P(i, 14): i in [1..n]]: n in [1..40]]; // Bruno Berselli, Apr 18 2016

[(n+1)*(n+2)*(n+3)*(n+4)*(12*n+5)/120: n in [0..40]]; // Bruno Berselli, Apr 18 2016

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {1, 17, 87, 287, 742, 1638}, 40]
Table[(n + 1) (n + 2) (n + 3) (n + 4) (12 n + 5)/120, {n, 0, 40}]

O.g.f.: (1 + 11*x)/(1 - x)^6.
E.g.f.: (120 + 1920*x + 3240*x^2 + 1520*x^3 + 245*x^4 + 12*x^5)*exp(x)/120.
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).
a(n) = (n + 1)*(n + 2)*(n + 3)*(n + 4)*(12*n + 5)/120.
Sum_{n>=0} 1/a(n) = 20*((15552*(6*log(2) + 3*log(3) + 2*sqrt(3)*log(2 - sqrt(3)) + (2 - sqrt(3))*Pi) - 29449)/531867) = 1.07654258697...

Edited by Bruno Berselli, Apr 18 2016

cp's OEIS Frontend

A005585 5-dimensional pyramidal numbers: a(n) = n*(n+1)*(n+2)*(n+3)*(2n+3)/5!.

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A034827 a(n) = 2*binomial(n,4).

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A093563 (6,1)-Pascal triangle.

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A027810 a(n) = (n+1)*binomial(n+5, 5).

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A290775 Number of 5-cycles in the n-triangular honeycomb bishop graph.

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A290779 Number of 6-cycles in the n-triangular honeycomb bishop graph.

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A005585 5-dimensional pyramidal numbers: a(n) = n(n+1)(n+2)(n+3)(2n+3)/5!.

A136526 Coefficients polynomials B(x, n) = ((1 + a + b)x - c)B(x, n-1) - abB(x, n-2) with a = 3, b = 2, and c = 0.