A002414 -id:A002414 - cp's OEIS frontend

A202689 a(n) = (2n)! * (n+1)! / 2^(2n).

1, 1, 9, 270, 18900, 2551500, 589396500, 214540326000, 115851776040000, 88626608670600000, 92614806060777000000, 128364121200236922000000, 230285233433225038068000000, 523898906060586961604700000000, 1485253398681764036149324500000000

Offset: 0

Alexander R. Povolotsky, Dec 22 2011

a(n) is always an integer since a(n+1)/a(n) = n^3 + 7/2*n^2 + 7/2*n + 1 which is always an integer. [D. S. McNeil, Dec 22 2011]

To further follow the above comment from D. S. McNeil: a(n+1)/a(n) is given in A002414. [Alexander R. Povolotsky, Dec 23 2011]

Vincenzo Librandi, Table of n, a(n) for n = 0..100
T. Piezas, Notes and conjectures on properties of polynomials, arising in "Construction Of Binomial Sums For π And Polylogarithmic Constants Inspired by BBP Formulas" (by Boris Gourévitch, Jesús Guillera Goyanes) and also relevant to further unpublished follow-up work by J. Cullen, T. Piezas, J. Guillera, and B. Gourevitch.
Index to divisibility sequences

Cf. A002414.

[Factorial(2*n)*Factorial(n+1)/2^(2*n): n in [0..15]]; // Vincenzo Librandi, Feb 09 2012

Table[(2n)!(n+1)!/2^(2n),{n,0,20}] (* Vincenzo Librandi, Feb 09 2012 *)

a(n)=(2*n)!*(n+1)!>>(2*n) \\ Charles R Greathouse IV, Dec 23 2011

a(n) = (2n)!(n+1)! / 2^(2n).
a(n+1) = a(n)*(n^3 + 7/2*n^2 + 7/2*n + 1).
a(n+1) = a(n)*A002414(n+1) for n >= 0.

A218326 Odd octagonal pyramidal numbers.

Original entry on oeis.org

1, 9, 135, 231, 765, 1045, 2275, 2835, 5049, 5985, 9471, 10879, 15925, 17901, 24795, 27435, 36465, 39865, 51319, 55575, 69741, 74949, 92115, 98371, 118825, 126225, 150255, 158895, 186789, 196765, 228811, 240219, 276705, 289641, 330855, 345415, 391645, 407925

Offset: 1

Ant King, Oct 27 2012

nonn

			The sequence of octagonal pyramidal numbers A002414 begins 1, 9, 30, 70, 135, 231, 364, 540, 765, 1045, … As the third odd term is 135, then a(3) = 135.

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

Cf. A002414, A218327.

LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,9,135,231,765,1045,2275},38]

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) + 384.
a(n) = (4*n-(-1)^n-1)*(4*n-(-1)^n-3)*(4*n-(-1)^n-4)/8.
G. f. x(1+8*x+123*x^2+72*x^3+159*x^4+16*x^5+5*x^6)/((1-x)^4*(1+x)^3).

A241747 Triangle read by rows: T(n,k) = (4n+3)(4*k+3).

Original entry on oeis.org

9, 21, 49, 33, 77, 121, 45, 105, 165, 225, 57, 133, 209, 285, 361, 69, 161, 253, 345, 437, 529, 81, 189, 297, 405, 513, 621, 729, 93, 217, 341, 465, 589, 713, 837, 961, 105, 245, 385, 525, 665, 805, 945, 1085, 1225, 117, 273, 429, 585, 741, 897, 1053, 1209, 1365, 1521

Offset: 0

Vincenzo Librandi, Apr 29 2014

A016838(n) first diagonal.
A085027(n) second diagonal.
A017629(n) column k=0.
Row sums give the second bisection of A002414: 9, 70, 231, 540, 1045, 1794, 2835, 4216, ... [Bruno Berselli, May 08 2014]

			Triangle begins:
n\k |   0     1     2    3    4    5     6     7     8     9
----|--------------------------------------------------------
0   |   9;
1   |  21,   49;
2   |  33,   77,  121;
3   |  45,  105,  165, 225;
4   |  57,  133,  209, 285, 361;
5   |  69,  161,  253, 345, 437, 529;
6   |  81,  189,  297, 405, 513, 621,  729;
7   |  93,  217,  341, 465, 589, 713,  837,  961;
8   | 105,  245,  385, 525, 665, 805,  945, 1085, 1225;
9   | 117,  273,  429, 585, 741, 897, 1053, 1209, 1365, 1521;
.....

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A002145, A004767, A016838, A017629, A085027.

[(4*n+3)*(4*k+3): k in [0..n], n in [0..15]]; /* or, as triangle: */ [[(4*n+3)*(4*k+3): k in [0..n]]: n in [0..10]];

t[n_, k_] := (4 n + 3) (4 k + 3); Table[t[n, k], {n, 0, 10}, {k, n}] // Flatten

Edited by Alois P. Heinz and Bruno Berselli, May 08 2014

A269429 Alternating sum of octagonal pyramidal numbers.

Original entry on oeis.org

0, -1, 8, -22, 48, -87, 144, -220, 320, -445, 600, -786, 1008, -1267, 1568, -1912, 2304, -2745, 3240, -3790, 4400, -5071, 5808, -6612, 7488, -8437, 9464, -10570, 11760, -13035, 14400, -15856, 17408, -19057, 20808, -22662, 24624, -26695, 28880, -31180, 33600

Offset: 0

Ilya Gutkovskiy, Feb 26 2016

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Pyramidal Number
Index entries for linear recurrences with constant coefficients, signature (-3,-2,2,3,1).

Cf. A000292, A002414, A002717, A173196, A266677.

Table[((2 n^3 + 4 n^2 - 1) (-1)^n + 1)/4, {n, 0, 40}]
LinearRecurrence[{-3, -2, 2, 3, 1}, {0, -1, 8, -22, 48}, 41]

a(n)=(2*n^3 + 4*n^2 - 1)*(-1)^n\/4 \\ Charles R Greathouse IV, Jul 26 2016

G.f.: x*(1 - 5*x)/((x - 1)*(x + 1)^4).
a(n) = ((2*n^3 + 4*n^2 - 1)*(-1)^n + 1)/4.
a(n) = Sum_{k = 0..n} (-1)^k*A002414(k).
Sum_{n>=1} 1/a(n) = -0.906890389180715042293808708467278316660747358... . - Vaclav Kotesovec, Feb 26 2016

A290055 Expansion of x(1 + 4x + x^2)/((1 - x)^5*(1 + x)^4).

Original entry on oeis.org

0, 1, 5, 10, 26, 40, 80, 110, 190, 245, 385, 476, 700, 840, 1176, 1380, 1860, 2145, 2805, 3190, 4070, 4576, 5720, 6370, 7826, 8645, 10465, 11480, 13720, 14960, 17680, 19176, 22440, 24225, 28101, 30210, 34770, 37240, 42560, 45430, 51590, 54901, 61985, 65780, 73876, 78200, 87400, 92300, 102700, 108225, 119925, 126126

Offset: 0

Ilya Gutkovskiy, Aug 15 2017

More generally, the generalized 4-dimensional figurate numbers are convolution of the sequence {1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...} with generalized polygonal numbers (A195152).

Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A001082, A002414, A002419, A002624, A027656, A195152, A287143.

CoefficientList[Series[x (1 + 4 x + x^2)/((1 - x)^5 (1 + x)^4), {x, 0, 51}], x]
LinearRecurrence[{1, 4, -4, -6, 6, 4, -4, -1, 1}, {0, 1, 5, 10, 26, 40, 80, 110, 190}, 52]

x='x+O('x^99); concat(0, Vec(x*(1+4*x+x^2)/((1-x)^5*(1 + x)^4))) \\ Altug Alkan, Aug 15 2017

G.f.: x*(1 + 4*x + x^2)/((1 - x)^5*(1 + x)^4).
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9).
Generalized 4-dimensional figurate numbers (A002419): (3*n - 1)*binomial(n + 2,3)/2, n = 0,+1,-3,+2,-4,+3,-5, ...
Convolution of the sequences A027656 and A001082 (with offset 0).
a(n) = (2*n+3+(-1)^n)*(2*n+7+(-1)^n)*(6*n^2+30*n+5-(2*n+5)*(-1)^n)/1536. - Luce ETIENNE, Nov 18 2017

A338496 Least number of octagonal pyramidal numbers needed to represent n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 3, 4, 5, 1, 2, 3, 4, 5, 6, 4, 5, 6, 2, 3, 4, 5, 6, 7, 5, 6, 7, 3, 4, 5, 6, 7, 8, 6, 7, 8, 4, 5, 6, 2, 3, 4, 5, 6, 7, 5, 6, 7, 3, 1, 2, 3, 4, 5, 6, 7, 8, 4, 2, 3, 4, 5, 6, 7, 8, 9, 5, 3, 4, 3, 4, 5, 6, 7, 8, 6, 4, 5, 4, 2, 3

Offset: 1

Ilya Gutkovskiy, Oct 31 2020

nonn

Eric Weisstein's World of Mathematics, Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A002414, A104246, A338458, A338481, A338493, A338494, A338495.

A361226 Square array T(n,k) = k((1+2n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 2, 5, 3, 0, 3, 9, 12, 6, 0, 4, 13, 21, 22, 10, 0, 5, 17, 30, 38, 35, 15, 0, 6, 21, 39, 54, 60, 51, 21, 0, 7, 25, 48, 70, 85, 87, 70, 28, 0, 8, 29, 57, 86, 110, 123, 119, 92, 36, 0, 9, 33, 66, 102, 135, 159, 168, 156, 117, 45

Offset: 0

Paul Curtz, Mar 05 2023

The main diagonal is A002414.
The first upper diagonal is A160378(n+1).
The antidiagonals sums are A034827(n+2).
b(n) = (A034827(n+3) = 0, 2, 10, 30, 70, ...) - (A002414(n) = 0, 1, 9, 30, 70, ...) = 0, 1, 1, 0, 0, 5, 21, 56, ... .
b(n+2) = A299120(n). b(n+4) = A033275(n). b(n+4) - b(n) = A002492(n).

			The rows are
  0, 0,  1,  3,  6,  10,  15,  21, ...   = A161680
  0, 1,  5, 12, 22,  35,  51,  70, ...   = A000326
  0, 2,  9, 21, 38,  60,  87, 119, ...   = A005476
  0, 3, 13, 30, 54,  85, 123, 168, ...   = A022264
  0, 4, 17, 39, 70, 110, 159, 217, ...   = A022266
  ... .
Columns: A000004, A001477, A016813, A017197=3*A016777, 2*A017101, 5*A016873, 3*A017581, 7*A017017, ... (coefficients from A026741).
Difference between two consecutive rows: A000290. Hence A143844.
This square array read by antidiagonals leads to the triangle
  0
  0   0
  0   1   1
  0   2   5   3
  0   3   9  12   6
  0   4  13  21  22  10
  0   5  17  30  38  35  15
  ... .

Cf. A000004, A000290, A000326, A001477, A002414, A005476, A016777, A016813, A016873, A017017, A017101, A017197, A017581, A022264, A022266, A026741, A034827, A160378, A161680, A360962.

Cf. A002492, A033275, A143844, A299120.

T[n_, k_] := k*((2*n + 1)*k - 1)/2; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 05 2023 *)

a(n) = { my(row = (sqrtint(8*n+1)-1)\2, column = n - binomial(row + 1, 2)); binomial(column, 2) + column^2 * (row - column) } \\ David A. Corneth, Mar 05 2023

# Seen as a triangle:
from functools import cache
@cache
def Trow(n: int) -> list[int]:
    if n == 0: return [0]
    r = Trow(n - 1)
    return [r[k] + k * k if k < n else r[n - 1] + n - 1 for k in range(n + 1)]
for n in range(7): print(Trow(n)) # Peter Luschny, Mar 05 2023

Take successively sequences n*(n-1)/2, n*(3*n-1)/2, n*(5*n-1)/2, ... listed in the EXAMPLE section.
G.f.: y*(x + y)/((1 - y)^3*(1 - x)^2). - Stefano Spezia, Mar 06 2023
E.g.f.: exp(x+y)*y*(2*x + y + 2*x*y)/2. - Stefano Spezia, Feb 21 2024

A363288 a(n) = (2n^3 - n^2 + 3n - 2)/2.

Original entry on oeis.org

1, 8, 26, 61, 119, 206, 328, 491, 701, 964, 1286, 1673, 2131, 2666, 3284, 3991, 4793, 5696, 6706, 7829, 9071, 10438, 11936, 13571, 15349, 17276, 19358, 21601, 24011, 26594, 29356, 32303, 35441, 38776, 42314, 46061, 50023, 54206, 58616, 63259, 68141, 73268, 78646, 84281

Offset: 1

Wesley Ivan Hurt, May 25 2023

For n >= 3, a(n) is the sum of all multiples of n XOR n-1 that are <= n^2.

Winston de Greef, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217, A002411, A002414, A046092, A162264.

[(2*n^3 - n^2 + 3*n - 2)/2 : n in [1..50]];

Table[(2 n^3 - n^2 + 3 n - 2)/2, {n, 100}]
LinearRecurrence[{4, -6, 4, -1}, {1, 8, 26, 61}, 50]

a(n) = n^3 - 1 + (-n^2 + 3*n)/2 \\ Winston de Greef, Jun 01 2023

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = A002411(n) + (n-1)*A000217(n+1) - A046092(n-1).
a(n) = A162264(n-1) + 1 for n >= 2. - Hugo Pfoertner, Jun 02 2023
G.f.: x*(1 - 5*x + 2*x^2 - 4*x^3)/(1 - x)^4. - Stefano Spezia, Jun 03 2023

A378918 Indices of octagonal pyramidal numbers that are both octagonal and octagonal pyramidal.

Original entry on oeis.org

0, 1, 10, 18, 49785, 91839

Offset: 1

Kelvin Voskuijl, Dec 10 2024

			10 is a term because the 10th octagonal pyramidal number (1045) is also the 19th octagonal number.

Cf. A000567 (octagonal numbers), A002414 (octagonal pyramidal numbers).
Cf. A344376.
Cf. A378361 (octagonal indices).

A344376(n) = A002414(a(n)).

A386206 Triangle read by rows: T(n,k) = n^2 - k, with 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 4, 3, 2, 9, 8, 7, 6, 16, 15, 14, 13, 12, 25, 24, 23, 22, 21, 20, 36, 35, 34, 33, 32, 31, 30, 49, 48, 47, 46, 45, 44, 43, 42, 64, 63, 62, 61, 60, 59, 58, 57, 56, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90

Offset: 0

Stefano Spezia, Jul 15 2025

			The triangle begins as:
   0;
   1,  0;
   4,  3,  2;
   9,  8,  7,  6;
  16, 15, 14, 13, 12;
  25, 24, 23, 22, 21, 20;
  36, 35, 34, 33, 32, 31, 30;
  49, 48, 47, 46, 45, 44, 43, 42;
  64, 63, 62, 61, 60, 59, 58, 57, 56;
  ...

Vincenzo Librandi, Table of n, a(n) for n = 0..3320 (rows 0..80)

Cf. A000290 (k=0), A002414 (row sums), A005563, A008865, A028347 (k=4), A028872 (k=3), A028875 (k=5), A279019 (diagonal).

[[n^2-k: k in [0..n]]: n in [0..9]]; // Vincenzo Librandi, Jul 17 2025

T[n_,k_]:=n^2-k; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

G.f.: x*(1 + x + 2*x*y^2 + 5*x^3*y^2 - x^2*y*(4 + 5*y))/((1 - x)^3*(1 - x*y)^3).
T(n,1) = A005563(n-1) for n > 0.
T(n,2) = A008865(n) for n > 1.

cp's OEIS Frontend

A202689 a(n) = (2n)! * (n+1)! / 2^(2n).

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A218326 Odd octagonal pyramidal numbers.

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A241747 Triangle read by rows: T(n,k) = (4*n+3)*(4*k+3).

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A269429 Alternating sum of octagonal pyramidal numbers.

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A290055 Expansion of x*(1 + 4*x + x^2)/((1 - x)^5*(1 + x)^4).

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A338496 Least number of octagonal pyramidal numbers needed to represent n.

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A361226 Square array T(n,k) = k*((1+2*n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

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A363288 a(n) = (2*n^3 - n^2 + 3*n - 2)/2.

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A241747 Triangle read by rows: T(n,k) = (4n+3)(4*k+3).

A290055 Expansion of x(1 + 4x + x^2)/((1 - x)^5*(1 + x)^4).

A361226 Square array T(n,k) = k((1+2n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

A363288 a(n) = (2n^3 - n^2 + 3n - 2)/2.