A002414 -id:A002414 - cp's OEIS frontend

A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-4+3h, n>=1, h>=1, and ** = convolution.

2, 9, 5, 24, 18, 8, 50, 42, 27, 11, 90, 80, 60, 36, 14, 147, 135, 110, 78, 45, 17, 224, 210, 180, 140, 96, 54, 20, 324, 308, 273, 225, 170, 114, 63, 23, 450, 432, 392, 336, 270, 200, 132, 72, 26, 605, 585, 540, 476, 399, 315

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A213820.
Antidiagonal sums: A153978.
Row 1, (1,2,3,4,...)**(2,5,8,11,...): A006002.
Row 2, (1,2,3,4,...)**(5,8,11,14,...): is it the sequence A212343?.
Row 3, (1,2,3,4,...)**(8,11,14,17,...): (k^3 + 8*k^2 + 7*k)/2.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
2....9....24....50....90....147
5....18...42....80....135...210
8....27...60....110...180...273
11...36...78....140...225...336
14...45...96....170...270...399
17...54...114...200...315...462

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A212500

b[n_]:=n;c[n_]:=3n-1;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213819 *)
Table[t[n,n],{n,1,40}] (* A213820 *)
d/2 (* A002414 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A153978 *)
s1/2 (* A001296 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x(3*n-1 - (3*n-4)*x) and g(x) = (1-x)^4.

A261720 Array of pyramidal (3-dimensional figurate numbers) read by antidiagonals.

Original entry on oeis.org

1, 1, 4, 1, 5, 10, 1, 6, 14, 20, 1, 7, 18, 30, 35, 1, 8, 22, 40, 55, 56, 1, 9, 26, 50, 75, 91, 84, 1, 10, 30, 60, 95, 126, 140, 120, 1, 11, 34, 70, 115, 161, 196, 204, 165, 1, 12, 38, 80, 135, 196, 252, 288, 285, 220, 1, 13, 42, 90, 155, 231, 308, 372, 405, 385, 286

Offset: 1

Gary W. Adamson, Aug 29 2015

First few sequences in the array:
  1,  4, 10, 20,  35,  56,  84, 120, 165,  220, ... A000292
  1,  5, 14, 30,  55,  91, 140, 204, 285,  385, ... A000330
  1,  6, 18, 40,  75, 126, 196, 288, 405,  550, ... A002411
  1,  7, 22, 50,  95, 161, 252, 372, 525,  715, ... A002412
  1,  8, 26, 60, 115, 196, 308, 456, 645,  880, ... A002413
  1,  9, 30, 70, 135, 231, 364, 540, 765, 1045, ... A002414
  1, 10, 34, 80, 155, 266, 420, 624, 885, 1210, ... A007584
  ...
The corresponding bases to rows are: Triangle, Square, Pentagon, Hexagon, Heptagon, Octagon, ...

			Row 2: (1, 5, 14, 30, 55, ...) = (1, 4, 10, 20, 35, ...) + (0, 1, 4, 10, 20, 35, ...).
(1, 7, 22, 50, ...) is the binomial transform of (1, 6, 9, 4, 0, 0, 0, ...) 3rd row in Pascal's triangle (1,4) followed by zeros. (1, 7, 22, 50, ...) is the third partial sum of (1, 4, 4, 4, ...).

Albert H. Beiler, "Recreations in the Theory of Numbers"; Dover, 1966, p. 194.

Cf. A000292, A000330, A002411, A002412, A002413, A002414, A007584, A220084.

Similar to A080851 but without row n=0.

T[n_,k_]:=k(k+1)((k-1)n+3)/6; Flatten[Table[T[n-k+1,k],{n,11},{k,n}]] (* Stefano Spezia, Aug 15 2024 *)

T(n,k) = A080851(n,k).
Given: first sequence in the array is A000292: (1, 4, 10, 20, 35, ...) Subsequent rows are generated by adding (0, 1, 4, 10, 20, 35, ...) to the current row.
n-th row is the binomial transform of row 3 in Pascal's triangle (1,n) followed by zeros.  Alternatively, begin with (1, 4, 10, 20, ...) being the binomial transform of (1, 3, 3, 1, 0, 0, 0, ...). Add (0, 1, 2, 1, 0, 0, 0, ...) to the latter to obtain the inverse binomial transform of the next row: (1, 5, 14, 30, 55,..); then repeat the operation.
The row starting (1, N, ...) is the 3rd partial sum of (1, (N-3), (N-3), (N-3), ...).
From Stefano Spezia, Aug 15 2024: (Start)
T(n,k) = k*(k + 1)*((k - 1)*n + 3)/6.
G.f. as array: x*y*(1 + x*(y - 1))/((1 - x)^2*(1 - y)^4).
E.g.f. as array: exp(y)*y*(exp(x)*(6 + 3*(1 + x)*y + x*y^2) - 3*(2 + y))/6. (End)

A328910 Number of nontrivial solutions to Erdős's Last Equation in n variables, x_1...x_n = n*(x_1 + ... + x_n), with 1 <= x_1 <= ... <= x_n.

Original entry on oeis.org

2, 6, 8, 8, 17, 14, 19, 27, 25, 15, 33, 16, 30, 43, 45, 18, 55, 24, 43, 55, 43, 22, 92, 43, 35, 68, 69, 25, 107, 34, 80, 56, 48, 61, 130, 32, 45, 65, 119, 29, 113, 36, 72, 154, 64, 27, 178, 69, 87, 74, 85, 24, 197, 97, 145, 91, 51, 37, 182, 39, 54, 189, 203, 82, 173

Offset: 2

M. F. Hasler, Nov 07 2019

nonn

For n = 1, any (x_1) would be a solution, therefore the offset is 2.
For any n, the equation would also admit the trivial zero solution (0,...,0). If any x_k = 0, then all x_i must be zero, so there is no other solution in the nonnegative integers.
This sequence gives row sums of A328911 which lists the number of solutions with given number of components > 1.
The Shiu paper gives different values for f(2), f(3) and f(8) than those given here, but the author has confirmed (personal communication) that 2, 6 and 19 are correct.
The multiset [1^(n-2),n+1,n*(2*n-1)] is the solution with maximum product. - Hugo Pfoertner, Nov 08 2019
From David A. Corneth, Nov 08 2019: (Start)
Given x_1,...,x_{n-1}, one can find the value x_n by solving the resulting linear univariate equation.
For example, for n = 4, if we are given (1, 1, 7, x_4) then we can solve 4*(9 + x_4) = 7*x_4, getting 36 = 3*x_4, i.e., x_4 = 12. As 12 is an integer and >= x_3 = 7, we have a new solution: (1, 1, 7, 12). (End)
In any solution, we have x_1*...*x_{n-1} <= n^2, implying that a(n) is finite for all n > 1. Furthermore, x_1 = x_2 = ... = x_k = 1 for k = n - 1 - floor(2*log_2(n)). - Max Alekseyev, Nov 10 2019

			For n = 2 variables, we have the equation x1*x2 = 2*(x1 + x2) with positive integer solutions (3,6) and (4,4).
For n = 3, we have 3 solutions in C_1(3) = {(1, 4, 15), (1, 5, 9), (1, 6, 7)} (with 2 components > 1), and 3 others in C_2(3) = {(2, 2, 12), (2, 3, 5), (3,3,3)} (with 3 components > 1), for a total of a(3) = 6.
For n = 4 we have the 8 solutions (1, 1, 5, 28), (1, 1, 6, 16), (1, 1, 7, 12), (1, 1, 8, 10), (1, 2, 3, 12), (1, 2, 4, 7), (1, 3, 4, 4) and (2, 2, 2, 6).
For n = 5, the solutions are, omitting initial components x_i = 1: {(6, 45), (7, 25), (9, 15), (10, 13), (2, 3, 35), (2, 5, 9), (3, 3, 10), (3, 5, 5)}.
For n = 6, the solutions are (omitting x_i = 1): {(7, 66), (8, 36), (9, 26), (10, 21), (11, 18), (12, 16), (2, 4, 27), (2, 5, 15), (2, 6, 11), (2, 7, 9), (3, 3, 18), (3, 4, 10), (3, 6, 6), (2, 2, 2, 24), (2, 2, 3, 9), (2, 2, 4, 6), (2, 3, 3, 5)}.
For n = 9, the 27 solutions are (omitting '1's): {(10, 153), (11, 81), (12, 57), (13, 45), (15, 33), (17, 27), (18, 25), (21, 21), (2, 5, 117), (2, 6, 42), (2, 7, 27), (2, 9, 17), (2, 12, 12), (3, 4, 39), (3, 5, 21), (3, 6, 15), (3, 7, 12), (3, 9, 9), (4, 4, 18), (5, 5, 9), (6, 6, 6), (2, 2, 3, 36), (2, 2, 6, 9), (2, 3, 3, 13), (3, 3, 3, 7), (3, 3, 4, 5), (2, 3, 3, 3, 3)}.

Lars Blomberg, Posting to the Sequence Fans Mailing List, Nov 07 2019, seems to have been the first person to notice that there were problems with the published values (given in A328980). - N. J. A. Sloane, Nov 08 2019
R. K. Guy, Sum equals product. in: Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, chapter D24, (2004), 299-301 (citing Erdős's question dated Aug 19, 1996)

David A. Corneth, Table of n, a(n) for n = 2..6250
David A. Corneth, List of solutions and product of variables of solutions for n = 2..160, omitting ones
Peter Shiu, On Erdős's Last Equation, Amer. Math. Monthly, 126 (2019), 802-808; correction, 127:5 (2020), 478.

Row sums of A328911.

Cf. A000384, A002414, A213820 (apparently maximum product occurring in solutions), A328980, A329205, A329206.

a(n,show=1)={my(s=0,d);forvec(x=vector(n-1,i,[1,n\(sqrt(2)-1)]), 0<(d=vecprod(x)-n) && n*vecsum(x)%d==0 && n*vecsum(x)\d >= x[n-1] &&s++ &&show &&printf("%d,",concat(x,n*vecsum(x)\d)),1);s}

{ A328910(n,k=n-1,m=n^2,p=1,s=0,y=1) = if(k==0, return( p>n && Mod(n*s,p-n)==0 && n*s>=(p-n)*y ) ); sum(x=y, sqrtnint(m,k), A328910(n,k-1,m\x,p*x,s+x,x) ); } \\ Max Alekseyev, Nov 10 2019

More terms from David A. Corneth, Nov 07 2019

A152015 a(n) = n^3 - n^2 - n.

Original entry on oeis.org

0, -1, 2, 15, 44, 95, 174, 287, 440, 639, 890, 1199, 1572, 2015, 2534, 3135, 3824, 4607, 5490, 6479, 7580, 8799, 10142, 11615, 13224, 14975, 16874, 18927, 21140, 23519, 26070, 28799, 31712, 34815, 38114, 41615, 45324, 49247, 53390, 57759, 62360

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 20 2008

For n > 1, these are also the largest positive integers k such that k + n divides k^3 + n^2. For all n > 1 and p > 1, the largest positive integer k such that k + n divides k^p + n^(p-1) is given by k = n^p - (-n)^(p-1) - n. Here, p = 3. - Derek Orr, Aug 13 2014

Derek Orr, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002414, A005449.

[n^3-n^2-n : n in [0..50]]; // Wesley Ivan Hurt, Aug 13 2014

a:=n->sum(-1+sum(1+sum(1,i=2..n),j=2..n),k=1..n): seq(a(n), n=0..44); # Zerinvary Lajos, Dec 22 2008
A152015:=n->n^3-n^2-n: seq(A152015(k), k=0..100); # Wesley Ivan Hurt, Oct 06 2013

lst={};Do[AppendTo[lst,n^3-n^2-n],{n,0,5!}];lst
Table[n^3-n^2-n, {n,0,100}] (* Wesley Ivan Hurt, Oct 06 2013 *)
LinearRecurrence[{4,-6,4,-1},{0,-1,2,15},50] (* Harvey P. Dale, Sep 08 2024 *)

vector(100, n, (n-1)^3-(n-1)^2-(n-1)) \\ Derek Orr, Aug 13 2014

G.f.: -x*(1-6*x-x^2)/(1-x)^4. - Bruno Berselli, Jul 27 2012
a(n) = A002414(n) - A005449(n). - Wesley Ivan Hurt, Oct 06 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Aug 13 2014
E.g.f.: exp(x)*x*(x^2 + 2*x - 1). - Stefano Spezia, Apr 15 2022

Offset changed by Bruno Berselli, Jul 27 2012

A279224 Expansion of Product_{k>=1} 1/(1 - x^(k(k+1)(2*k-1)/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 16, 16, 17, 17, 17, 18, 18, 18, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 24, 24, 26, 26, 26, 27, 27, 27, 28, 29, 29, 31, 32

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Number of partitions of n into nonzero octagonal pyramidal numbers (A002414).

			a(10) = 2 because we have [9, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Index to sequences related to pyramidal numbers
Index entries for related partition-counting sequences

Cf. A002414, A068980, A279220, A279221, A279222, A279223.

nmax=100; CoefficientList[Series[Product[1/(1 - x^(k (k + 1) (2 k - 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^(k*(k+1)*(2*k-1)/2)).

A366017 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 5 * A(x)).

Original entry on oeis.org

0, 1, 9, 132, 2365, 47169, 1005564, 22431720, 517122117, 12222124035, 294569159313, 7212098118888, 178877944712844, 4484938858752940, 113488477622130600, 2894560146756466320, 74335973069605120725, 1920587845828953301479, 49886703842977713177723, 1301959618949870922531300, 34123873581608909988904245

Offset: 0

Ilya Gutkovskiy, Sep 26 2023

nonn

Reversion of g.f. for octagonal pyramidal numbers (with signs).

Eric Weisstein's World of Mathematics, Series Reversion

Cf. A002293, A002414, A064089, A365754, A365818, A366014, A366015, A366016.

nmax = 20; A[] = 0; Do[A[x] = x (1 + A[x])^4/(1 - 5 A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
CoefficientList[InverseSeries[Series[x (1 - 5 x)/(1 + x)^4, {x, 0, 20}], x], x]	
Join[{0}, Table[1/n Sum[Binomial[n + k - 1, k] Binomial[4 n, n - k - 1] 5^k, {k, 0, n - 1}], {n, 1, 20}]]

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(4*n,n-k-1) * 5^k for n > 0.

A294845 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(2*k-1)/2).

Original entry on oeis.org

1, 1, 9, 39, 136, 511, 1785, 6139, 20404, 66406, 211418, 660752, 2030172, 6139231, 18300573, 53823451, 156344596, 448886205, 1274840165, 3583595734, 9976530997, 27520998775, 75262394273, 204130567402, 549318633095, 1467178746342, 3890697051314, 10246833932820, 26809705578787, 69702402930045

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the octagonal pyramidal numbers (A002414).

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A002414, A258343, A281156, A294838, A294841, A294842, A294843, A294844.

nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1) (2 k - 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (d + 1) (2 d - 1)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 29}]

G.f.: Product_{k>=1} (1 + x^k)^A002414(k).

a(n) ~ exp(-2401 * Pi^16 / (2267481600000000 * Zeta(5)^3) - 49*Pi^8 * Zeta(3) / (388800000 * Zeta(5)^2) - Zeta(3)^2 / (400*Zeta(5)) + (343*Pi^12 / (87480000000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (18000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) - (49*Pi^8 / (6480000 * 2^(3/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + 3^(1/5)*Zeta(3) / (2^(13/5) * (5*Zeta(5))^(2/5))) * n^(2/5) + (7*Pi^4 / (1080 * 2^(2/5) * 3^(1/5) * (5*Zeta(5))^(3/5))) * n^(3/5) + (3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) / 2^(11/5)) * n^(4/5)) * 3^(1/5) * Zeta(5)^(1/10) / (2^(11/20) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - Vaclav Kotesovec, Nov 10 2017

A329757 Doubly octagonal pyramidal numbers.

Original entry on oeis.org

0, 1, 765, 27435, 345415, 2469420, 12352956, 48294610, 157609530, 447989355, 1141711615, 2663460261, 5775482505, 11777133550, 22789550070, 42150245460, 74946834916, 128723876325, 214401953745, 347453633935, 549386792955, 849592039296, 1287617552320, 1915941609990, 2803320397950, 4038796372975

Offset: 0

Ilya Gutkovskiy, Nov 20 2019

Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A000567, A002414, A140236, A264892, A329753, A329754, A329755, A329756.

A002414[n_] := n (n + 1) (2 n - 1)/2; a[n_] := A002414[A002414[n]]; Table[a[n], {n, 0, 25}]
Table[Sum[k (3 k - 2), {k, 0, n (n + 1) (2 n - 1)/2}], {n, 0, 25}]
nmax = 25; CoefficientList[Series[x (1 + 755 x + 19830 x^2 + 105370 x^3 + 158255 x^4 + 70629 x^5 + 7930 x^6 + 110 x^7)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 765, 27435, 345415, 2469420, 12352956, 48294610, 157609530, 447989355}, 26]

G.f.: x*(1 + 755*x + 19830*x^2 + 105370*x^3 + 158255*x^4 + 70629*x^5 + 7930*x^6 + 110*x^7)/(1 - x)^10.
a(n) = A002414(A002414(n)).
a(n) = Sum_{k=0..A002414(n)} A000567(k).
a(n) = n *(2*n-1) *(n+1) *(2*n^3+n^2-n+2) *(2*n^3+n^2-n-1) /8 . - R. J. Mathar, Nov 28 2019

A038758 Number of ways of covering a 2n X 2n lattice by 2n^2 dominoes with exactly 4 horizontal (or vertical) dominoes.

Original entry on oeis.org

16, 281, 1785, 7175, 22015, 56406, 126966, 259170, 490050, 871255, 1472471, 2385201, 3726905, 5645500, 8324220, 11986836, 16903236, 23395365, 31843525, 42693035, 56461251, 73744946, 95228050, 121689750, 154012950, 193193091

Offset: 2

Yong Kong (ykong(AT)curagen.com), May 06 2000

			a(3) = 281 because we have 281 ways to cover a 4 X 4 lattice with exactly 4 horizontal dominoes and exactly 14 vertical dominoes.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical Review, 124 (1961), 1664-1672.
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A004003, A002414, A054344.

[(1/24)*n*(n-1)*(n+1)*(12*n^3-11*n-10): n in [2..30]]; // Vincenzo Librandi, Oct 22 2013

CoefficientList[Series[(16 + 169 x + 154 x^2 + 21 x^3)/(1 - x)^7, {x, 0, 30}], x] (* Vincenzo Librandi, Oct 22 2013 *)

a(n) = (1/24)*n*(n-1)*(n+1)*(12*n^3-11*n-10).

G.f.: x^2*(16+169*x+154*x^2+21*x^3)/(1-x)^7. [Colin Barker, Jun 26 2012]

More terms from James Sellers, May 10 2000

A054344 Number of ways of covering a 2n X 2n lattice with 2n^2 dominoes of which exactly 6 are horizontal (or vertical) dominoes.

Original entry on oeis.org

9, 1064, 21656, 197484, 1143366, 4927524, 17240292, 51631617, 137044523, 330284988, 735542444, 1533609350, 3024043008, 5684167992, 10249533240, 17821214019, 30006185613, 49097892704, 78305096016

Offset: 2

Yong Kong (ykong(AT)curagen.com), May 06 2000

			a(3) = 1064 because we have 1064 ways to cover a 36 X 36 lattice with exactly 6 horizontal (or vertical) dominoes and exactly 12 vertical (or horizontal) dominoes.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical Review, 124 (1961), 1664-1672.
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A004003, A002414, A038758.

CoefficientList[Series[(x^9-10*x^8+45*x^7-36*x^6+3096*x^5 +17256*x^4 +27724*x^3+11421*x^2+974*x+9)/(x-1)^10,{x,0,30}],x] (* Vincenzo Librandi, Jun 26 2012 *)

a(n) = (1/720)*n*(n+1)*(120*n^7-300*n^6-70*n^5+363*n^4+416*n^3-231*n^2-394*n-264).

G.f.: x^2*(x^9-10*x^8+45*x^7-36*x^6+3096*x^5+17256*x^4+27724*x^3+11421*x^2+974*x+9)/(x-1)^10. - Colin Barker, Jun 26 2012

cp's OEIS Frontend

A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3*n-4+3*h, n>=1, h>=1, and ** = convolution.

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A261720 Array of pyramidal (3-dimensional figurate numbers) read by antidiagonals.

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A328910 Number of nontrivial solutions to Erdős's Last Equation in n variables, x_1*...*x_n = n*(x_1 + ... + x_n), with 1 <= x_1 <= ... <= x_n.

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PARI

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A152015 a(n) = n^3 - n^2 - n.

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A279224 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)*(2*k-1)/2)).

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A366017 G.f. A(x) satisfies: A(x) = x * (1 + A(x))^4 / (1 - 5 * A(x)).

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A294845 Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(2*k-1)/2).

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A213819 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = 3n-4+3h, n>=1, h>=1, and ** = convolution.

A328910 Number of nontrivial solutions to Erdős's Last Equation in n variables, x_1...x_n = n*(x_1 + ... + x_n), with 1 <= x_1 <= ... <= x_n.

A279224 Expansion of Product_{k>=1} 1/(1 - x^(k(k+1)(2*k-1)/2)).

A294845 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(2*k-1)/2).