A002412 -id:A002412 - cp's OEIS frontend

A323663 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is Sum_{j=1..n} binomial(j*k, k).

1, 1, 3, 1, 7, 6, 1, 21, 22, 10, 1, 71, 105, 50, 15, 1, 253, 566, 325, 95, 21, 1, 925, 3256, 2386, 780, 161, 28, 1, 3433, 19489, 18760, 7231, 1596, 252, 36, 1, 12871, 119713, 154085, 71890, 17857, 2926, 372, 45, 1, 48621, 748342, 1303753, 747860, 214396, 38332, 4950, 525, 55

Offset: 1

Seiichi Manyama, Jan 23 2019

			Square array begins:
    1,   1,    1,     1,       1,        1, ...
    3,   7,   21,    71,     253,      925, ...
    6,  22,  105,   566,    3256,    19489, ...
   10,  50,  325,  2386,   18760,   154085, ...
   15,  95,  780,  7231,   71890,   747860, ...
   21, 161, 1596, 17857,  214396,  2695652, ...
   28, 252, 2926, 38332,  539028,  7941438, ...
   36, 372, 4950, 74292, 1197036, 20212950, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns 1-3 give A000217, A002412, A116689.
Rows 1-3 give A000012, A244174, A029848.
Main diagonal is A096131.
Cf. A060539.

A372751 a(n) = (3n^5 + 4n^3 - n)/6.

Original entry on oeis.org

1, 21, 139, 554, 1645, 4031, 8631, 16724, 30009, 50665, 81411, 125566, 187109, 270739, 381935, 527016, 713201, 948669, 1242619, 1605330, 2048221, 2583911, 3226279, 3990524, 4893225, 5952401, 7187571, 8619814, 10271829, 12167995, 14334431, 16799056, 19591649

Offset: 1

Kelvin Voskuijl, May 12 2024

Sums of hexagonal numbers (A000384) in successive groups of length 1,2,3,etc, so 1, 6+15, 28+45+66, 91+120+153+190, etc.

			The hexagonal numbers and their groups summed begin
  1, 6, 15, 28, 45, 66, 91, 120, 153, 190
  \/ \---/  \--------/  \---------------/
  1,   21,     139,            554

Paolo Xausa, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000384 (hexagonal numbers), A002412 (their partial sums).

Cf. A260513 (for triangular numbers), A072474 (for squares), A372583 (for pentagonal numbers), A075664 (cubes).

A372751[n_] := (3*n^5 + 4*n^3 - n)/6; Array[A372751, 50] (* Paolo Xausa, Jun 19 2024 *)

From Stefano Spezia, May 12 2024: (Start)
G.f.: x*(1 + 15*x + 28*x^2 + 15*x^3 + x^4)/(1 - x)^6.
E.g.f.: exp(x)*x*(6 + 57*x + 79*x^2 + 30*x^3 + 3*x^4)/6. (End)

A374194 a(n) is the smallest number which can be represented as the sum of two nonzero hexagonal pyramidal numbers in exactly n ways, or -1 if no such number exists.

Original entry on oeis.org

2, 5972, 5170425

Offset: 1

Ilya Gutkovskiy, Jun 30 2024

There are no further positive terms <= 10^15. - Michael S. Branicky, Jun 30 2024

			a(2) = 5972 = 1222 + 4750 = 1925 + 4047.

Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number

Cf. A002412, A011541, A334013, A374141, A374168, A374193.

A083215 a(n) = 1 + Sum(prime(i)(2i-1): 1<=i<=n).

Original entry on oeis.org

3, 12, 37, 86, 185, 328, 549, 834, 1225, 1776, 2427, 3278, 4303, 5464, 6827, 8470, 10417, 12552, 15031, 17800, 20793, 24190, 27925, 32108, 36861, 42012, 47471, 53356, 59569, 66236, 73983, 82236, 91141, 100454, 110735, 121456, 132917

Offset: 1

Reinhard Zumkeller, Jun 01 2003

nonn

Eric Weisstein's World of Mathematics, Figurate Number.

Cf. A002412.

Equals 1 + A316322(n).

nxt[{n_, a_}] := {n + 1, a + Prime[n + 1] (2 n + 1)}; NestList[nxt,{1,2},50][[All,2]]+1 (* Harvey P. Dale, Jul 05 2018 *)

a(n) = 1 + sum(i=1, n, prime(i)*(2*i-1)); \\ Michel Marcus, Jan 22 2022

Definition corrected by Harvey P. Dale, Jul 03 2018

A095802 Upper right triangular matrix T^2, where T(i,j) = (-1)^i(1-2i) for 1 <= i <= j.

Original entry on oeis.org

1, -2, 9, 3, -6, 25, -4, 15, -10, 49, 5, -12, 35, -14, 81, -6, 21, -20, 63, -18, 121, 7, -18, 45, -28, 99, -22, 169, -8, 27, -30, 77, -36, 143, -26, 225, 9, -24, 55, -42, 117, -44, 195, -30, 289, -10, 33, -40, 91, -54, 165, -52, 255, -34, 361, 11, -30, 65, -56, 135, -66, 221, -60, 323, -38, 441, -12, 39, -50, 105, -72, 187, -78, 285, -68, 399, -42, 529

Offset: 1

Gary W. Adamson, Jun 07 2004

Equivalently, (lower left) triangle M^2 = transpose(T)^2. The following description refers to the lower triangular version, but OEIS's "TABL" link displays the values more appropriately as an upper right triangle. - M. F. Hasler, Apr 18 2009

For n rows, use matrices in each row from the sequence 1, -3, 5, -7, ... (filling in with zeros except for the n-th row). Let the matrix = M, then square and delete the zeros. For example, the 3-row generator would be [1 0 0 / 1 -3 0 / 1 -3 5] = M. The nonzero elements of M^2 give the first 6 terms of the sequence.

			The matrix
  [  1   0   0   0 ...]
  [  1  -3   0   0 ...]
  [  1  -3   5   0 ...]
  [  1  -3   5  -7 ...]
squared yields
  [ +1   0   0   0 ...]
  [ -2  +9   0   0 ...]
  [ +3  -6  25   0 ...]
  [ -4  15 -10  49 ...]; the lower left triangle gives this sequence: 1; -2, 9; 3, -6, 25; ...

Row sums with signs as shown = A002412, Hexagonal pyramidal numbers: (1, 7, 22, 50, 95, ...).

Cf. A002412.

T=matrix(12,12,i,j,if(j>=i,(-1)^i*(1-2*i)))^2; concat(vector(#T,i,vecextract(T[,i],2^i-1))) \\ M. F. Hasler, Apr 18 2009

Diagonal elements are the odd squares: a(k(k+1)/2)=(2k+1)^2. First element in row k is (-1)^k*k. - M. F. Hasler, Apr 18 2009

Edited and extended by M. F. Hasler, Apr 18 2009

A103217 Hexagonal numbers triangle read by rows: T(n,k)=(n+1-k)(2(n+1-k)-1).

Original entry on oeis.org

1, 6, 1, 15, 6, 1, 28, 15, 6, 1, 45, 28, 15, 6, 1, 66, 45, 28, 15, 6, 1, 91, 66, 45, 28, 15, 6, 1, 120, 91, 66, 45, 28, 15, 6, 1, 153, 120, 91, 66, 45, 28, 15, 6, 1, 190, 153, 120, 91, 66, 45, 28, 15, 6, 1, 231, 190, 153, 120, 91, 66, 45, 28, 15, 6, 1, 276, 231, 190, 153, 120, 91, 66

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 25 2005

The triangle is generated by the product A*B = B*A of the infinite lower triangular matrices A =
   1 0 0 0...
   1 1 0 0...
   1 1 1 0...
   1 1 1 1...
   ...
  and B =
    1 0 0 0...
    5 1 0 0...
    9 5 1 0...
   13 9 5 1...
  ...
The only prime hexagonal pyramidal number is 7. The only semiprime hexagonal pyramidal numbers are: 22, 95, 161. All greater hexagonal pyramidal numbers A002412 have at least 3 prime factors. Note that 7337 = 11 * 23 * 29 is a palindromic 3-brilliant number and 65941 = 23 * 47 * 61 is 3-brilliant. - Jonathan Vos Post, Jan 26 2005

			Triangle begins:
  1,
  6,1,
  15,6,1,
  28,15,6,1,
  45,28,15,6,1,
  66,45,28,15,6,1,
  91,66,45,28,15,6,1,

Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
Eric Weisstein's World of Mathematics, Hexagonal Number.
Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number.

Row sums give A002412 (hexagonal pyramidal numbers).

Cf. A000384, A002412.

T[n_, k_] := (n + 1 - k)*(2*(n + 1 - k) - 1); Flatten[ Table[ T[n, k], {n, 0, 10}, {k, 0, n}]] (* Robert G. Wilson v, Feb 10 2005 *)

T(n, k) = (n+1-k)*(2*(n+1-k)-1); for(i=0,10, for(j=0,i,print1(T(i,j),",")); print())

A103218 Triangle read by rows: T(n, k) = (2k+1)(n+1-k)^2.

Original entry on oeis.org

1, 4, 3, 9, 12, 5, 16, 27, 20, 7, 25, 48, 45, 28, 9, 36, 75, 80, 63, 36, 11, 49, 108, 125, 112, 81, 44, 13, 64, 147, 180, 175, 144, 99, 52, 15, 81, 192, 245, 252, 225, 176, 117, 60, 17, 100, 243, 320, 343, 324, 275, 208, 135, 68, 19, 121, 300, 405, 448, 441, 396, 325, 240

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 25 2005

The triangle is generated from the product A * B of the infinite lower triangular matrix A =
0 0 0...
1 0 0...
3 1 0...
5 3 1...
... and B =
0 0 0...
3 0 0...
3 5 0...
3 5 7...
...

			Triangle begins:
1,
4,3,
9,12,5,
16,27,20,7,
25,48,45,28,9,

Row sums give A002412 (hexagonal pyramidal numbers).
T(n, 0)=A000290(n+1) (the squares);
T(n, 1)=3*n^2=A033428(n);
T(n, 2)=5*n^2=A033429(n+1);
T(n, 3)=7*n^2=A033582(n+2);
Cf. A103219 (product B*A), A002412, A000290.

T[n_, k_] := (2*k + 1)*(n + 1 - k)^2; Flatten[ Table[ T[n, k], {n, 0, 10}, {k, 0, n}]] (* Robert G. Wilson v, Feb 10 2005 *)

T(n, k) = (2*k+1)*(n+1-k)^2; for(i=0,10, for(j=0,i,print1(T(i,j),","));print())

A130269 A002260 * A051340.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 11, 13, 16, 15, 16, 18, 21, 25, 21, 22, 24, 27, 31, 36, 28, 29, 31, 34, 38, 43, 49, 36, 37, 39, 42, 46, 51, 57, 64, 45, 46, 48, 51, 55, 60, 66, 73, 81, 55, 56, 58, 61, 65, 70, 76, 83, 91, 100

Offset: 1

Gary W. Adamson, May 18 2007

Row sums = A002412: (1, 7, 22, 50, 95, ...).

			First few rows of the triangle:
   1;
   3,  4;
   6,  7,  9;
  10, 11, 13, 16;
  15, 16, 18, 21, 25,
  21, 22, 24, 27, 31, 36;
  28, 29, 31, 34, 38, 43, 49;
  ...

Cf. A002260, A051340, A002412.

A002260 * A051340 as infinite lower triangular matrices.

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

A144080 Positions of hexagonal pyramidal numbers in the EKG sequence.

Original entry on oeis.org

1, 14, 19, 46, 85, 155, 233, 350, 494, 659, 878, 1163, 1460, 1828, 2260, 2726, 3271, 3873, 4564, 5321, 6151, 7032, 8044, 9151

Offset: 1

Parthasarathy Nambi, Sep 09 2008

nonn

			The hexagonal pyramidal number 9500 is located at position 9151 in the EKG sequence.

A002412, A064413

cp's OEIS Frontend

A323663 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is Sum_{j=1..n} binomial(j*k, k).

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A372751 a(n) = (3*n^5 + 4*n^3 - n)/6.

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A374194 a(n) is the smallest number which can be represented as the sum of two nonzero hexagonal pyramidal numbers in exactly n ways, or -1 if no such number exists.

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A083215 a(n) = 1 + Sum(prime(i)*(2*i-1): 1<=i<=n).

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A095802 Upper right triangular matrix T^2, where T(i,j) = (-1)^i*(1-2*i) for 1 <= i <= j.

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A103217 Hexagonal numbers triangle read by rows: T(n,k)=(n+1-k)*(2*(n+1-k)-1).

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A103218 Triangle read by rows: T(n, k) = (2*k+1)*(n+1-k)^2.

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A130269 A002260 * A051340.

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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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A372751 a(n) = (3n^5 + 4n^3 - n)/6.

A083215 a(n) = 1 + Sum(prime(i)(2i-1): 1<=i<=n).

A095802 Upper right triangular matrix T^2, where T(i,j) = (-1)^i(1-2i) for 1 <= i <= j.

A103217 Hexagonal numbers triangle read by rows: T(n,k)=(n+1-k)(2(n+1-k)-1).

A103218 Triangle read by rows: T(n, k) = (2k+1)(n+1-k)^2.