A002413 -id:A002413 - cp's OEIS frontend

A279223 Expansion of Product_{k>=1} 1/(1 - x^(k(k+1)(5*k-2)/6)).

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 16, 16, 16, 16, 17, 17, 18, 18, 20, 20, 20, 20, 21, 21, 22, 22, 24, 24, 25, 25, 26, 26, 27, 27, 29, 29, 31, 31, 32, 32, 33, 33, 35, 35, 37, 37

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Number of partitions of n into nonzero heptagonal pyramidal numbers (A002413).

			a(9) = 2 because we have [8, 1] and [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]
Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number
Index to sequences related to pyramidal numbers
Index entries for related partition-counting sequences

Cf. A002413, A068980, A279220, A279221, A279222, A279224.

nmax=95; CoefficientList[Series[Product[1/(1 - x^(k (k + 1) (5 k - 2)/6)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^(k*(k+1)*(5*k-2)/6)).

A134081 Triangle T(n, k) = binomial(n, k)((2k+1)*(n-k) +k+1)/(k+1), read by rows.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 12, 8, 1, 5, 22, 26, 11, 1, 6, 35, 60, 45, 14, 1, 7, 51, 115, 125, 69, 17, 1, 8, 70, 196, 280, 224, 98, 20, 1, 9, 92, 308, 546, 574, 364, 132, 23, 1, 10, 117, 456, 966, 1260, 1050, 552, 171, 26, 1

Offset: 0

Gary W. Adamson, Oct 07 2007

			First few rows of the triangle are:
  1;
  2,  1;
  3,  5,   1;
  4, 12,   8,   1;
  5, 22,  26,  11,  1;
  6, 35,  60,  45, 14,  1;
  7, 51, 115, 125, 69, 17, 1;
  ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A002064, A007318, A112295.

Columns: A000027, A000326, A002413, A051740, A051879.

A134081:= func< n,k | Binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1) >;
[A134081(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2021

T[n_, k_]:= Binomial[n, k]*((2*k+1)*(n-k) +k+1)/(k+1);
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 17 2021 *)

def A134081(n,k): return binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1)
flatten([[A134081(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 17 2021

Binomial transform of A112295(unsigned).
From G. C. Greubel, Feb 17 2021: (Start)
T(n, k) = binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1).
Sum_{k=0..n} T(n, k) = 2^n *n + 1 = A002064(n). (End)

A294844 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(5*k-2)/6).

Original entry on oeis.org

1, 1, 8, 34, 114, 411, 1380, 4573, 14650, 45995, 141296, 426364, 1265443, 3698011, 10657134, 30312395, 85183177, 236681860, 650686538, 1771098691, 4775571943, 12762628737, 33821018537, 88909273699, 231945942992, 600700301298, 1544897610261, 3946762859175, 10018454809275, 25274880698255

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the heptagonal pyramidal numbers (A002413).

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
Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number

Cf. A002413, A258343, A281156, A294837, A294840, A294842, A294843, A294845.

nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1) (5 k - 2)/6), {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) (5 d - 2)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 29}]

G.f.: Product_{k>=1} (1 + x^k)^A002413(k).

a(n) ~ (3*Zeta(5))^(1/10) / (2^(479/720) * 5^(3/10) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (1312200000000000 * Zeta(5)^3) - 49 * Pi^8 * Zeta(3) / (405000000 * Zeta(5)^2) - Zeta(3)^2 / (750*Zeta(5)) + (343*Pi^12 / (60750000000 * 2^(3/5) * 3^(1/5) * 5^(2/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (22500 * 2^(3/5) * 3^(1/5) * 5^(2/5) * Zeta(5)^(6/5))) * n^(1/5) - (49*Pi^8 / (5400000 * 2^(1/5) * 3^(2/5) * 5^(4/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(6/5) * 5^(4/5) * (3*Zeta(5))^(2/5))) * n^(2/5) + (7*Pi^4 / (900 * 2^(4/5) * 5^(1/5) * (3*Zeta(5))^(3/5))) * n^(3/5) + (5^(7/5) * (3*Zeta(5))^(1/5) / 2^(12/5)) * n^(4/5)). - Vaclav Kotesovec, Nov 10 2017

A329756 Doubly heptagonal pyramidal numbers.

Original entry on oeis.org

0, 1, 456, 14976, 181780, 1273970, 6293756, 24395756, 79119496, 223821235, 568280240, 1321714636, 2858876956, 5817509516, 11237224740, 20751835560, 36849296016, 63215722181, 105182448536, 170297734920, 269047574180, 415753060646, 629674964556, 936359517556

Offset: 0

Ilya Gutkovskiy, Nov 20 2019

Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number
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. A000566, A002413, A140236, A264891, A329753, A329754, A329755, A329757.

A002413[n_] := n (n + 1) (5 n - 2)/6; a[n_] := A002413[A002413[n]]; Table[a[n], {n, 0, 25}]
Table[Sum[k (5 k - 3)/2, {k, 0, n (n + 1) (5 n - 2)/6}], {n, 0, 25}]
nmax = 25; CoefficientList[Series[x (1 + 446 x + 10461 x^2 + 52420 x^3 + 75580 x^4 + 32544 x^5 + 3504 x^6 + 44 x^7)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 456, 14976, 181780, 1273970, 6293756, 24395756, 79119496, 223821235}, 26]

G.f.: x*(1 + 446*x + 10461*x^2 + 52420*x^3 + 75580*x^4 + 32544*x^5 + 3504*x^6 + 44*x^7)/(1 - x)^10.
a(n) = A002413(A002413(n)).
a(n) = Sum_{k=0..A002413(n)} A000566(k).
a(n) = n *(5*n-2) *(n+1) *(5*n^3+3*n^2-2*n+6) *(25*n^3+15*n^2-10*n-12)/1296. - R. J. Mathar, Nov 28 2019

A125234 Triangle T(n,k) read by rows: the k-th column contains the k-fold iterated partial sum of A000566.

Original entry on oeis.org

1, 7, 1, 18, 8, 1, 34, 26, 9, 1, 55, 60, 35, 10, 1, 81, 115, 95, 45, 11, 1, 112, 196, 210, 140, 56, 12, 1, 148, 308, 406, 350, 196, 68, 13, 1, 189, 456, 714, 756, 546, 264, 81, 14, 1, 235, 645, 1170, 1470, 1302, 810, 345, 95, 15, 1, 286, 880, 1815, 2640, 2772, 2112, 1155, 440, 110, 16, 1

Offset: 1

Gary W. Adamson, Nov 24 2006

The leftmost column contains the heptagonal numbers A000566.
The adjacent columns to the right are A002413, A002418, A027800, A051946, A050484.
Row sums = 1, 8, 27, 70, 161, 348, 727, ... = 6*(2^n-1)-5*n.

			First few rows of the triangle are:
  1;
  7, 1;
  18, 8, 1;
  34, 26, 9, 1;
  55, 60, 35, 10, 1;
  81, 115, 95, 45, 11, 1;
  112, 196, 210, 140, 56, 12, 1;
Example: T(6,2) = 95 = 35 + 60 = T(5,2) + T(5,1).

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

Cf. A000566, A002413, A002418, A027800, A051946, A050484.

Analogous triangles for the hexagonal and pentagonal numbers are A125233 and A125232.

A000566 := proc(n) n*(5*n-3)/2 ; end: A125234 := proc(n,k) if k = 0 then A000566(n); elif k>= n then 0 ; else procname(n-1,k-1)+procname(n-1,k) ; fi; end: seq(seq(A125234(n,k),k=0..n-1),n=1..16) ; # R. J. Mathar, Sep 09 2009

A000566[n_] := n(5n-3)/2;
T[n_, k_] := Which[k == 0, A000566[n], k >= n, 0, True, T[n-1, k-1] + T[n-1, k] ];
Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Oct 26 2023, after R. J. Mathar *)

T(n,0) = A000566(n). T(n,k) = T(n-1,k) + T(n-1,k-1), k>0.

Edited and extended by R. J. Mathar, Sep 09 2009

A211796 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k<=x^k+y^k.

Original entry on oeis.org

1, 8, 1, 26, 7, 1, 60, 22, 7, 1, 115, 51, 22, 7, 1, 196, 99, 50, 22, 7, 1, 308, 168, 96, 50, 22, 7, 1, 456, 265, 163, 95, 50, 22, 7, 1, 645, 393, 255, 161, 95, 50, 22, 7, 1, 880, 556, 378, 253, 161, 95, 50, 22, 7, 1, 1166, 760, 534, 374, 252, 161, 95, 50, 22, 7

Offset: 1

Clark Kimberling, Apr 21 2012

Row 1:  A002413
Row 2:  A211634
Row 3:  A211650
Limiting row sequence: A002412
Let R be the array in A211796 and let R' be the array in A211799.  Then R(k,n)+R'(k,n)=3^(n-1).
See the Comments at A211790.

			Northwest corner:
1...8...26...60...115...196...308
1...7...22...51...99....168...265
1...7...22...50...96....163...255
1...7...22...50...95....161...253
1...7...22...50...95....161...252

Cf. A211790.

z = 48;
t[k_, n_] := Module[{s = 0},
   (Do[If[w^k <= x^k + y^k, s = s + 1],
       {w, 1, #}, {x, 1, #}, {y, 1, #}] &[n]; s)];
Table[t[1, n], {n, 1, z}]  (* A002413 *)
Table[t[2, n], {n, 1, z}]  (* A211634 *)
Table[t[3, n], {n, 1, z}]  (* A211650 *)
TableForm[Table[t[k, n], {k, 1, 12}, {n, 1, 16}]]
Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]] (* A211796 *)
Table[k (k - 1) (2 k - 1)/6, {k, 1,
  z}] (* row-limit sequence, A002412 *)
(* Peter J. C. Moses, Apr 13 2012 *)

A266085 Alternating sum of heptagonal numbers.

Original entry on oeis.org

0, -1, 6, -12, 22, -33, 48, -64, 84, -105, 130, -156, 186, -217, 252, -288, 328, -369, 414, -460, 510, -561, 616, -672, 732, -793, 858, -924, 994, -1065, 1140, -1216, 1296, -1377, 1462, -1548, 1638, -1729, 1824, -1920, 2020, -2121, 2226, -2332, 2442, -2553

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Heptagonal Number
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A000566, A002413, A006578, A008728, A035608, A083392, A089594.

Unsigned terms give antidiagonal sums of A204154. - Nathaniel J. Strout, Nov 14 2019

[((10*n^2+4*n-3)*(-1)^n+3)/8: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

R:=PowerSeriesRing(Integers(), 50); [0] cat  Coefficients(R!(-x*(1 - 4*x)/((1 - x)*(1 + x)^3))); // Marius A. Burtea, Nov 13 2019

Table[((10 n^2 + 4 n - 3) (-1)^n + 3)/8, {n, 0, 50}]
CoefficientList[Series[(x - 4 x^2)/(x^4 + 2 x^3 - 2 x - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)
LinearRecurrence[{-2,0,2,1},{0,-1,6,-12},60] (* Harvey P. Dale, Jan 26 2023 *)

x='x+O('x^100); concat(0, Vec(-x*(1-4*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

G.f.: -x*(1 - 4*x)/((1 - x)*(1 + x)^3).
a(n) = ((10*n^2 + 4*n - 3)*(-1)^n + 3)/8.
a(n) = Sum_{k = 0..n} (-1)^k*A000566(k).
Lim_{n -> infinity} a(n + 1)/a(n) = -1.
a(n) = (-1)^n*A008728(5*n-5) for n>0. - Bruno Berselli, Dec 21 2015
E.g.f.: (1/8)*exp(-x)*(-3 + 3*exp(2*x) - 14*x + 10*x^2). - Stefano Spezia, Nov 13 2019

A140729 Diagonal A(n,n) of array A(k,n) = Product of first n of k-gonal pyramidal numbers.

Original entry on oeis.org

40, 2100, 324000, 117771500, 86640153600, 115851776040000, 260111401804800000, 922852527136155000000, 4931966428685936640000000, 38193820496218904209973280000, 415101787718859995456102400000000

Offset: 3

Jonathan Vos Post, May 25 2008

The array A(k,n) = Product of first n k-gonal pyramidal numbers begins:
===================================================================
..|n=1|n=2|..n=3|...n=4..|......n=5....|......n=6......|......n=7......|.......n=8.........|
k=3|.1.|.4.|..40.|....800.|.......28000.|.......1568000.|.....131712000.|.......15805440000.|A087047
k=4|.1.|.5.|..70.|...2100.|......115500.|......10510500.|....1471470000.|......300179880000.|
k=5|.1.|.6.|.108.|...4320.|......324000.|......40824000.|....8001504000.|.....2304433152000.|
k=6|.1.|.7.|.154.|...7700.|......731500.|.....117771500.|...29678418000.|....11040371496000.|
k=7|.1.|.8.|.208.|..12480.|.....1435200.|.....281299200.|...86640153600.|....39507910041600.|
k=8|.1.|.9.|.270.|.718900.|.....2551500.|.....589396500.|..214540326000.|...115851776040000.|
===================================================================

			a(3) = product of the first 3 triangular pyramidal (tetrahedral) numbers (A000292) = A087047(3) = 1 * 4 * 10 = 40.
a(4) = product of the first 4 square pyramidal numbers (A000330) = 1 * 5 * 14 * 30 = 2100.
a(5) = product of the first 5 pentagonal pyramidal numbers (A002411) = 1 * 6 * 18 * 40 * 75 = 324000.
a(6) = product of the first 6 hexagonal pyramidal numbers (A002412) = 1 * 7 * 22 * 50 * 95 * 161 = 117771500.
a(7) = product of the first 7 heptagonal pyramidal numbers (A002413) = 1 * 8 * 26 * 60 * 115 * 196 * 308 = 86640153600.
a(8) = product of the first 8 octagonal pyramidal numbers (A002414) = 1 * 9 * 30 * 70 * 135 * 231 * 364 * 540 = 115851776040000.

Eric W. Weisstein, Pyramidal Number

Cf. A000292, A000330, A002411, A002412, A002413, A002414, A140729.

A130729 := proc(n) n!*(n+1)!*(n-2)^n*pochhammer(1+(5-n)/(n-2),n)/6^n ; end: seq(A130729(n),n=3..30) ; # R. J. Mathar, May 31 2008

A(k,n) = PRODUCT[j=1..n] (1/6)*j*(j+1)*[(k-2)*j+(5-k)].

a(n) ~ Pi^(3/2) * n^(4*n + 1/2) / (2^(n - 3/2) * 3^(n-1) * exp(3*n+2)) * (1 + (3*log(n) + 3*gamma + 5/4)/n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Aug 29 2023

More terms from R. J. Mathar, May 31 2008

A218324 Odd heptagonal pyramidal numbers.

Original entry on oeis.org

1, 115, 645, 1911, 4233, 7931, 13325, 20735, 30481, 42883, 58261, 76935, 99225, 125451, 155933, 190991, 230945, 276115, 326821, 383383, 446121, 515355, 591405, 674591, 765233, 863651, 970165, 1085095, 1208761, 1341483, 1483581, 1635375, 1797185, 1969331

Offset: 1

Ant King, Oct 26 2012

			The sequence of heptagonal pyramidal numbers A002413 begins 1, 8, 26, 60, 115, 196, 308, 456, 645, 880, ... . As the third odd term is 645, a(3) = 645.

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002413, A218325.

LinearRecurrence[{4,-6,4,-1},{1,115,645,1911},34]

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 320.
a(n) = (2*n-1)*(4*n-3)*(20*n-17)/3.
G.f.: x*(1 + 111*x + 191*x^2 + 17*x^3)/(1-x)^4.
E.g.f.: 17 + exp(x)*(160*x^3 + 144*x^2 + 54*x - 51)/3. - Elmo R. Oliveira, Aug 23 2025

A218325 Even heptagonal pyramidal numbers.

Original entry on oeis.org

8, 26, 60, 196, 308, 456, 880, 1166, 1508, 2380, 2920, 3536, 5016, 5890, 6860, 9108, 10396, 11800, 14976, 16758, 18676, 22940, 25296, 27808, 33320, 36330, 39516, 46436, 50180, 54120, 62608, 67166, 71940, 82156, 87608, 93296, 105400, 111826, 118508, 132660

Offset: 1

Ant King, Oct 26 2012

nonn

			The sequence of heptagonal pyramidal numbers A002413(n) begins 1, 8, 26, 60, 115, 196, 308, 456, 645, 880, … As the third even term is 60, then a(3) = 60.

Index entries for linear recurrences with constant coefficients, signature (1, 0, 3, -3, 0, -3, 3, 0, 1, -1).

Cf. A002413, A218324.

LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{8,26,60,196,308,456,880,1166,1508,2380},40]

a(n) = a(n-1) + 3*a(n-3) - 3*a(n-4) - 3*a(n-6) + 3*a(n-7) + a(n-9) - a(n-10).
a(n) = 3*a(n-3) - 3*a(n-6) + a(n-9) + 320.
a(n) = (phi(n)+3)*(phi(n)+12)(5*phi(n)-3)/4374, where phi(n) = 12*n - 3*cos(2*n*pi/3) + sqrt(3)*sin(2*n*pi/3).
G. f. 2*x*(4+9*x+17*x^2+56*x^3+29*x^4+23*x^5+20*x^6+2*x^7) / ((1-x)^4*(1+x+x^2)^3).

cp's OEIS Frontend

A279223 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)*(5*k-2)/6)).

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A134081 Triangle T(n, k) = binomial(n, k)*((2*k+1)*(n-k) +k+1)/(k+1), read by rows.

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A294844 Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(5*k-2)/6).

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A329756 Doubly heptagonal pyramidal numbers.

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A125234 Triangle T(n,k) read by rows: the k-th column contains the k-fold iterated partial sum of A000566.

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A211796 Rectangular array: R(k,n) = number of ordered triples (w,x,y) with all terms in {1,...,n} and w^k<=x^k+y^k.

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A266085 Alternating sum of heptagonal numbers.

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A140729 Diagonal A(n,n) of array A(k,n) = Product of first n of k-gonal pyramidal numbers.

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A279223 Expansion of Product_{k>=1} 1/(1 - x^(k(k+1)(5*k-2)/6)).

A134081 Triangle T(n, k) = binomial(n, k)((2k+1)*(n-k) +k+1)/(k+1), read by rows.

A294844 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(5*k-2)/6).