A002412 -id:A002412 - cp's OEIS frontend

A259844 Number A(n,k) of n X n upper triangular matrices (m_{i,j}) of nonnegative integers with k = Sum_{j=h..n} m_{h,j} - Sum_{i=1..h-1} m_{i,h} for all h in {1,...,n}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 7, 1, 1, 1, 4, 22, 40, 1, 1, 1, 5, 50, 351, 357, 1, 1, 1, 6, 95, 1686, 11275, 4820, 1, 1, 1, 7, 161, 5796, 138740, 689146, 96030, 1, 1, 1, 8, 252, 16072, 1010385, 25876312, 76718466, 2766572, 1

Offset: 0

Alois P. Heinz, Jul 06 2015

A(n,k) counts generalized Tesler matrices. For the definition of Tesler matrices see A008608.

			A(2,2) = 3: [1,1; 0,3], [2,0; 0,2], [0,2; 0,4].
Square array A(n,k) begins:
  1,   1,     1,      1,       1,       1, ...
  1,   1,     1,      1,       1,       1, ...
  1,   2,     3,      4,       5,       6, ...
  1,   7,    22,     50,      95,     161, ...
  1,  40,   351,   1686,    5796,   16072, ...
  1, 357, 11275, 138740, 1010385, 5244723, ...

Alois P. Heinz, Antidiagonals n = 0..15, flattened

Columns k=0-2 give: A000012, A008608, A259919.

Rows n=0+1,2-3 give: A000012, A000027(k+1), A002412(k+1).

b:= proc(n, i, l, k) option remember; (m-> `if`(m=0, 1,
      `if`(i=0, b(l[1]+k, m-1, subsop(1=NULL, l), k), add(
      b(n-j, i-1, subsop(i=l[i]+j, l), k), j=0..n))))(nops(l))
    end:
A:= (n, k)-> b(k, n-1, [0$(n-1)], k):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, i_, l_List, k_] := b[n, i, l, k] = Function[{m}, If[m == 0, 1, If[i == 0, b[l[[1]] + k, m-1, ReplacePart[l, 1 -> Sequence[]], k], Sum[b[n-j, i-1, ReplacePart[l, i -> l[[i]]+j], k], {j, 0, n}]]]][Length[l]]; A[n_, k_] := b[k, n-1, Array[0&, n-1], k]; A[0, ] = A[, 0] = 1; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

A294843 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(4*k-1)/6).

Original entry on oeis.org

1, 1, 7, 29, 93, 320, 1026, 3256, 9995, 30102, 88722, 257042, 732876, 2058370, 5703858, 15606076, 42203027, 112882223, 298849221, 783574536, 2035876825, 5244191462, 13398463986, 33967008194, 85476285603, 213583335753, 530099612487, 1307195997381, 3203555001240, 7804386224233

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the hexagonal pyramidal numbers (A002412).

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, Hexagonal Pyramidal Number

Cf. A002412, A028377, A258343, A281156, A294836, A294842, A294844, A294845.

nmax = 29; CoefficientList[Series[Product[(1 + x^k)^(k (k + 1)(4 k - 1)/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)(4 d - 1)/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)^A002412(k).

a(n) ~ exp(-2401 * Pi^16 / (671846400000000 * Zeta(5)^3) - 49*Pi^8 * Zeta(3) / (518400000 * Zeta(5)^2) - Zeta(3)^2 / (2400*Zeta(5)) + (343 * Pi^12 / (77760000000 * 15^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4*Zeta(3) / (72000 * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) - (49*Pi^8 / (8640000 * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (8 * (15*Zeta(5))^(2/5))) * n^(2/5) + (7*Pi^4 / (720 * (15*Zeta(5))^(3/5))) * n^(3/5) + (5*(15*Zeta(5))^(1/5)/4) * n^(4/5)) * (3*Zeta(5))^(1/10) / (2^(173/360) * 5^(2/5) * sqrt(Pi) * n^(3/5)). - Vaclav Kotesovec, Nov 10 2017

A329755 Doubly hexagonal pyramidal numbers.

Original entry on oeis.org

0, 1, 252, 7337, 84575, 576080, 2795121, 10700382, 34388362, 96606475, 243939410, 564840991, 1217275137, 2469392562, 4757404575, 8765621740, 15534503236, 26603512517, 44196596312, 71459197125, 112756874195, 174046844356, 263335062397, 391232840362, 571628456750, 822490729775

Offset: 0

Ilya Gutkovskiy, Nov 20 2019

Eric Weisstein's World of Mathematics, Hexagonal 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. A000384, A002412, A063249, A140236, A329753, A329754, A329756, A329757.

A002412[n_] := n (n + 1) (4 n - 1)/6; a[n_] := A002412[A002412[n]]; Table[a[n], {n, 0, 25}]
Table[Sum[k (2 k - 1), {k, 0, n (n + 1) (4 n - 1)/6}], {n, 0, 25}]
nmax = 25; CoefficientList[Series[x (1 + 242 x + 4862 x^2 + 22425 x^3 + 30465 x^4 + 12424 x^5 + 1248 x^6 + 13 x^7)/(1 - x)^10, {x, 0, nmax}], x]
LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {0, 1, 252, 7337, 84575, 576080, 2795121, 10700382, 34388362, 96606475}, 26]

G.f.: x*(1 + 242*x + 4862*x^2 + 22425*x^3 + 30465*x^4 + 12424*x^5 + 1248*x^6 + 13*x^7)/(1 - x)^10.
a(n) = A002412(A002412(n)).
a(n) = Sum_{k=0..A002412(n)} A000384(k).
a(n) = n *(4*n-1) *(n+1) *(4*n^3+3*n^2-n+6) *(8*n^3+6*n^2-2*n-3) / 648 . - R. J. Mathar, Nov 28 2019

A374502 Hexagonal pyramidal numbers that are products of smaller hexagonal pyramidal numbers.

Original entry on oeis.org

1, 4750, 1926049000, 655578709500, 9126464328696330

Offset: 1

Pontus von Brömssen, Jul 09 2024

a(6) > 10^19 (if it exists). - Pontus von Brömssen, Jul 14 2024

			1 is a term because it is a hexagonal pyramidal number and equals the empty product.
4750 is a term because it is a hexagonal pyramidal number and equals the product of the hexagonal pyramidal numbers 50 and 95.
1926049000 is a term because it is a hexagonal pyramidal number and equals the product of the hexagonal pyramidal numbers 9500 and 202742.
655578709500 is a term because it is a hexagonal pyramidal number and equals the product of the hexagonal pyramidal numbers 50, 31746, and 413015.
9126464328696330 is a term because it is a hexagonal pyramidal number and equals the product of the hexagonal pyramidal numbers 413015 and 22097174022.

Row n=6 of A374498.

Cf. A002412, A374500, A374501.

a(5) from Michael S. Branicky, Jul 09 2024

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 *)

A211798 R(k,n) = Sum_{y=1..n} Sum_{x=1..n} floor((x^k + y^k)^(1/k)), square array read by descending antidiagonals.

Original entry on oeis.org

2, 12, 1, 36, 7, 1, 80, 23, 7, 1, 150, 54, 22, 7, 1, 252, 103, 51, 22, 7, 1, 392, 175, 97, 50, 22, 7, 1, 576, 276, 164, 95, 50, 22, 7, 1, 810, 409, 258, 162, 95, 50, 22, 7, 1, 1100, 579, 382, 254, 161, 95, 50, 22, 7, 1, 1452, 791, 541, 375, 253, 161, 95, 50, 22

Offset: 1

Clark Kimberling, Apr 26 2012

			Northwest corner:
  2  12  36  80 150 252 392
  1   7  23  54 103 175 276
  1   7  22  51  97 164 258
  1   7  22  50  95 162 254
  1   7  22  50  95 161 254
  1   7  22  50  95 161 253

Cf. A002411 ((1/2) * row 1), A002412 (limiting row), A211791 (row 2), A211792 (row 3).

f[x_, y_, k_] := f[x, y, k] = Floor[(x^k + y^k)^(1/k)]
t[k_, n_] := Sum[Sum[f[x, y, k], {x, 1, n}], {y, 1, n}]
Table[t[1, n], {n, 1, 45}]  (* 2*A002411 *)
Table[t[2, n], {n, 1, 45}]  (* A211791 *)
Table[t[3, n], {n, 1, 45}]  (* A211792 *)
TableForm[Table[t[k, n], {k, 1, 12},
                 {n, 1, 16}]] (* A211798 *)
Flatten[Table[t[k, n - k + 1], {n, 1, 12}, {k, 1, n}]]

R(k,n) = Sum_{y=1..n} Sum_{x=1..n} floor((x^k + y^k)^(1/k)).

Definition changed by Georg Fischer, Sep 10 2022

A015225 Odd hexagonal pyramidal numbers.

Original entry on oeis.org

1, 7, 95, 161, 525, 715, 1547, 1925, 3417, 4047, 6391, 7337, 10725, 12051, 16675, 18445, 24497, 26775, 34447, 37297, 46781, 50267, 61755, 65941, 79625, 84575, 100647, 106425, 125077, 131747, 153171, 160797, 185185, 193831, 221375, 231105

Offset: 1

Mohammad K. Azarian

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Intersection of A002412 (hexagonal pyramidal) and A005408 (odd numbers).

LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,7,95,161,525,715,1547},36] (* Ant King, Oct 25 2012 *)

a(n)=(8*n-2*(-1)^n-7)*(1+(-1)^n-4*n)*(3+(-1)^n-4*n)/24 \\ Charles R Greathouse IV, Jul 30 2016

Odd numbers of the form n*(n+1)*(4n-1)/6.
From Ant King, Oct 25 2012: (Start)
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) +256.
a(n) = (8*n-2*(-1)^n-7)*(1+(-1)^n-4*n)*(3+(-1)^n-4*n)/24.
G.f.: x*(1+6*x+85*x^2+48*x^3+103*x^4+10*x^5+3*x^6) / ((1-x)^4*(1+x)^3). (End)
E.g.f.: (1/6)*(-9 - 6*x - 24*x^2 + 18*exp(x) + ( - 9 + 12*x + 36*x^2 + 32*x^3)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 30 2016

More terms from Erich Friedman.

A101492 Triangle read by rows: T(n,k) = (n-k+1)(4k+1).

Original entry on oeis.org

1, 2, 5, 3, 10, 9, 4, 15, 18, 13, 5, 20, 27, 26, 17, 6, 25, 36, 39, 34, 21, 7, 30, 45, 52, 51, 42, 25, 8, 35, 54, 65, 68, 63, 50, 29, 9, 40, 63, 78, 85, 84, 75, 58, 33, 10, 45, 72, 91, 102, 105, 100, 87, 66, 37, 11, 50, 81, 104, 119, 126, 125, 116, 99, 74, 41, 12, 55, 90, 117

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 21 2005

The triangle is generated from the product A*B
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...
1 5 0 0...
1 5 9 0...
1 5 9 13...
...
T(n+0,0) = 1*n = A000027(n+1),
T(n+0,1) = 5*n = A008587(n),
T(n+1,2) = 9*n = A008591(n),
T(n+2,3) = 13*n = A008595(n),
so, for example,
T(n,n) = 4*n+1 = A016813(n),
T(n+1,n) = 8*n+2 = A017089(n),
T(n,0)*T(n,1)/10 = A000217(n) (triangular numbers),
T(n,n)*T(n,0) = A001107(n+1) (10-gonal numbers: 4*n^2 - 3*n),
T(n,n)*T(n,1)/5 = A007742(n).

Muniru A Asiru, Rows n=0..150 of triangle, flattened

Row sums give hexagonal pyramidal numbers A002412.

Cf. A101493 for product B*A, A002412.

Flat(List([0..11],n->List([0..n],k->(n+1-k)*(4*k+1)))); # Muniru A Asiru, Mar 07 2019

[[(n+1-k)*(4*k+1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Mar 07 2019

Flatten[Table[(n+1-k)(4k+1),{n,0,15},{k,0,n}]] (* Harvey P. Dale, Jun 09 2011 *)

T(n, k) = if(k>n,0,(n-k+1)*(4*k+1));
for(i=0,10, for(j=0,i,print1(T(i,j),", "));print())

[[(n-k+1)*(4*k+1) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 07 2019

A177890 15-gonal (or pentadecagonal) pyramidal numbers: a(n) = n(n+1)(13*n-10)/6.

Original entry on oeis.org

0, 1, 16, 58, 140, 275, 476, 756, 1128, 1605, 2200, 2926, 3796, 4823, 6020, 7400, 8976, 10761, 12768, 15010, 17500, 20251, 23276, 26588, 30200, 34125, 38376, 42966, 47908, 53215, 58900, 64976, 71456, 78353, 85680, 93450, 101676, 110371, 119548, 129220

Offset: 0

Bruno Berselli, Dec 14 2010

Also a(n) = (15-m)*A000292(n-1) + n*(n+1)*((m-2)*n - (m-5))/6 being n*(n+1)*((m-2)*n - (m-5))/6 a m-gonal pyramidal number (1 < m < 15). For m=6, a(n) = 9*A000292(n-1) + A002412(n).

Inverse binomial transform of this sequence: 0, 1, 14, 13, 0, 0 (0 continued).

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93 (thirteenth row of the table).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. similar sequences listed in A237616.

Cf. A000292, A002412, A051867.

List([0..40], n-> n*(n+1)*(13*n-10)/6); # G. C. Greubel, Aug 30 2019

I:=[0,1,16,58]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2) +4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 04 2012

[n*(n+1)*(13*n-10)/6: n in [0..40]]; // G. C. Greubel, Aug 30 2019

seq(n*(n+1)*(13*n-10)/6, n=0..40); # G. C. Greubel, Aug 30 2019

CoefficientList[Series[x*(1+12*x)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 04 2012 *)
Table[n*(n-1)*(13*n-23)/6, {n,40}] (* G. C. Greubel, Aug 30 2019 *)
LinearRecurrence[{4,-6,4,-1},{0,1,16,58},40] (* Harvey P. Dale, Dec 21 2022 *)

vector(40, n, n*(n-1)*(13*n-23)/6) \\ G. C. Greubel, Aug 30 2019

[n*(n+1)*(13*n-10)/6 for n in (0..40)] # G. C. Greubel, Aug 30 2019

G.f.: x*(1+12*x)/(1-x)^4.
a(n) = Sum_{i=0..n} A051867(i).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jul 04 2012
a(n) = Sum_{i=0..n-1} (n-i)*(13*i+1), with a(0)=0. - Bruno Berselli, Feb 10 2014
E.g.f.: x*(6 + 42*x + 13*x^2)*exp(x)/6. - G. C. Greubel, Aug 30 2019

A185958 Accumulation array of the array max{n,k}, by antidiagonals.

Original entry on oeis.org

1, 3, 3, 6, 7, 6, 10, 13, 13, 10, 15, 21, 22, 21, 15, 21, 31, 34, 34, 31, 21, 28, 43, 49, 50, 49, 43, 28, 36, 57, 67, 70, 70, 67, 57, 36, 45, 73, 88, 94, 95, 94, 88, 73, 45, 55, 91, 112, 122, 125, 125, 122, 112, 91, 55, 66, 111, 139, 154, 160, 161, 160, 154, 139, 111, 66, 78, 133, 169, 190, 200, 203, 203, 200, 190, 169, 133, 78, 91, 157, 202, 230, 245, 251, 252, 251, 245, 230, 202, 157, 91, 105, 183, 238, 274, 295, 305, 308, 308, 305, 295, 274, 238, 183, 105

Offset: 1

Clark Kimberling, Feb 07 2011

A member of the accumulation chain
... < A185917 < A051125 < A185958 < ...,
where A051125, written as a rectangular array M, is given by M(n,k)=max{n,k}.  See A144112 for the definition of accumulation array.
row 1: A000217
row 2: A002061
diag (1,7,...): A002412
diag (3,13,..): A016061
antidiagonal sums: A070893

			Northwest corner:
1....3....6....10....15
3....7....13...21....31
6....13...22...34....49
10...21...34...50....70

Cf. A144112, A051125.

A := proc(n,k) option remember; ## n >= 0 and k = 0 .. n
    if k < 0 or k > n then
        0
    elif n = 0 then
        1
    else
        A(n-1,k) + A(n-1,k-1) - A(n-2,k-1) + max(n-k+1,k+1)
    end if
end proc: # Yu-Sheng Chang, Jun 05 2020

From Yu-Sheng Chang, Jun 05 2020: (Start)
O.g.f.: F(z,v) = -(v^2*z^3+v*z^3-3*v*z^2+1)/((v*z^2-v*z-z+1)^2*(v*z^2-1)*(z-1)*(v*z-1)).
T(n,k) = [v^k] 1/2*n^2*(v^(n+2)+1)/(1-v)^2+1/2*n*(3*v^(n+3)-7*v^(n+2)+7*v-3)/(-1+v)^3-1/2*v*((1-v^(1/2))^4*(-1)^n+(1+v^(1/2))^4)*v^(1/2*n)/(1-v)^4+(6*v^2+6*v^(n+2)+v^(n+4)-3*v^(n+3)-3*v+1)/(1-v)^4.
(End)

cp's OEIS Frontend

A259844 Number A(n,k) of n X n upper triangular matrices (m_{i,j}) of nonnegative integers with k = Sum_{j=h..n} m_{h,j} - Sum_{i=1..h-1} m_{i,h} for all h in {1,...,n}; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A294843 Expansion of Product_{k>=1} (1 + x^k)^(k*(k+1)*(4*k-1)/6).

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A329755 Doubly hexagonal pyramidal numbers.

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A374502 Hexagonal pyramidal numbers that are products of smaller hexagonal pyramidal numbers.

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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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A211798 R(k,n) = Sum_{y=1..n} Sum_{x=1..n} floor((x^k + y^k)^(1/k)), square array read by descending antidiagonals.

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A015225 Odd hexagonal pyramidal numbers.

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PARI

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A101492 Triangle read by rows: T(n,k) = (n-k+1)*(4*k+1).

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A294843 Expansion of Product_{k>=1} (1 + x^k)^(k(k+1)(4*k-1)/6).

A101492 Triangle read by rows: T(n,k) = (n-k+1)(4k+1).

A177890 15-gonal (or pentadecagonal) pyramidal numbers: a(n) = n(n+1)(13*n-10)/6.