A195152 -id:A195152 - cp's OEIS frontend

A195825 Square array T(n,k) read by antidiagonals, n>=0, k>=1, which arises from a generalization of Euler's Pentagonal Number Theorem.

1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 2, 1, 1, 1, 7, 3, 1, 1, 1, 1, 11, 4, 2, 1, 1, 1, 1, 15, 5, 3, 1, 1, 1, 1, 1, 22, 7, 4, 2, 1, 1, 1, 1, 1, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1, 56, 16, 7, 4, 3, 1, 1, 1, 1, 1, 1, 1, 77, 21, 10, 4

Offset: 0

Omar E. Pol, Sep 24 2011

In the infinite square array the column k is related to the generalized m-gonal numbers, where m = k+4. For example: the first column is related to the generalized pentagonal numbers A001318. The second column is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on ... (see the program in which A195152 is a table of generalized m-gonal numbers).
In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041 (see below the first row of the table):
========================================================
.                                          Column k of
.                                          this square
.       Generalized   Triangle  Triangle   array A195825
k    m    m-gonal       "A"       "B"      [row sums of
.         numbers                          triangle "B"
.                                          with a(0)=1]
========================================================
1    5    A001318     A195310   A175003      A000041
2    6    A000217     A195826   A195836      A006950
3    7    A085787     A195827   A195837      A036820
4    8    A001082     A195828   A195838      A195848
5    9    A118277     A195829   A195839      A195849
6   10    A074377     A195830   A195840      A195850
7   11    A195160     A195831   A195841      A195851
8   12    A195162     A195832   A195842      A195852
9   13    A195313     A195833   A195843      A196933
10  14    A195818     A210944   A210954      A210964
...
It appears that column 2 of the square array is A006950.
It appears that column 3 of the square array is A036820.
Conjecture: if k is odd then column k contains (k+1)/2 plateaus whose levels are the first (k+1)/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 2. Otherwise, if k is even then column k contains k/2 plateaus whose levels are the first k/2 terms of A210843 and whose lengths are k+1, k-1, k-3, k-5, ... 3. The sequence A210843 gives the levels of the plateaus of column k, when k -> infinity. For the visualization of the plateaus see the graph of a column, for example see the graph of A210964. - Omar E. Pol, Jun 21 2012

			Array begins:
    1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
    1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
    2,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
    3,  2,  1,  1,  1,  1,  1,  1,  1,  1, ...
    5,  3,  2,  1,  1,  1,  1,  1,  1,  1, ...
    7,  4,  3,  2,  1,  1,  1,  1,  1,  1, ...
   11,  5,  4,  3,  2,  1,  1,  1,  1,  1, ...
   15,  7,  4,  4,  3,  2,  1,  1,  1,  1, ...
   22, 10,  5,  4,  4,  3,  2,  1,  1,  1, ...
   30, 13,  7,  4,  4,  4,  3,  2,  1,  1, ...
   42, 16, 10,  5,  4,  4,  4,  3,  2,  1, ...
   56, 21, 12,  7,  4,  4,  4,  4,  3,  2, ...
   77, 28, 14, 10,  5,  4,  4,  4,  4,  3, ...
  101, 35, 16, 12,  7,  4,  4,  4,  4,  4, ...
  135, 43, 21, 13, 10,  5,  4,  4,  4,  4, ...
  176, 55, 27, 14, 12,  7,  4,  4,  4,  4, ...
  ...
Column 1 is A000041 which starts: [1, 1], 2, 3, 5, 7, 11, ... The column contains only one plateau: [1, 1] which has level 1 and length 2.
Column 3 is A036820 which starts: [1, 1, 1, 1], 2, 3, [4, 4], 5, 7, 10, ... The column contains two plateaus: [1, 1, 1, 1], [4, 4], which have levels 1, 4 and lengths 4, 2.
Column 6 is A195850 which starts: [1, 1, 1, 1, 1, 1, 1], 2, 3, [4, 4, 4, 4, 4], 5, 7, 10, 12, [13, 13, 13], 14, 16, 21, ... The column contains three plateaus: [1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4], [13, 13, 13], which have levels 1, 4, 13 and lengths 7, 5, 3.

Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium
Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Columns (1-10): A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
For another version see A211970.
Cf. A057077, A195152, A210843.

Column k is asymptotic to exp(Pi*sqrt(2*n/(k+2))) / (8*sin(Pi/(k+2))*n). - Vaclav Kotesovec, Aug 14 2017

A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 3, 1, 1, 0, 0, 3, 2, 1, 0, 5, 2, 3, 3, 1, 0, 0, 5, 4, 3, 4, 1, 0, 7, 3, 5, 6, 3, 5, 1, 0, 0, 7, 6, 5, 8, 3, 6, 1, 0, 9, 4, 7, 9, 5, 10, 3, 7, 1, 0, 0, 9, 8, 7, 12, 5, 12, 3, 8, 1, 0, 11, 5, 9, 12, 7, 15, 5, 14, 3, 9, 1, 0, 0, 11, 10, 9, 16, 7

Offset: 0

Omar E. Pol, Sep 14 2011

Also square array T(n,k) read by antidiagonals in which column k lists the multiples of k and the odd numbers interleaved, n>=0, k>=0. Also square array T(n,k) read by antidiagonals in which if n is even then row n lists the multiples of (n/2), otherwise if n is odd then row n lists a constant sequence: the all n's sequence. Partial sums of the numbers of column k give the column k of A195152. Note that if k >= 1 then partial sums of the numbers of the column k give the generalized m-gonal numbers, where m = k + 4.

All columns are multiplicative. - Andrew Howroyd, Jul 23 2018

			Array begins:
.  0,   0,   0,   0,   0,   0,   0,   0,   0,   0,...
.  1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
.  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,...
.  3,   3,   3,   3,   3,   3,   3,   3,   3,   3,...
.  0,   2,   4,   6,   8,  10,  12,  14,  16,  18,...
.  5,   5,   5,   5,   5,   5,   5,   5,   5,   5,...
.  0,   3,   6,   9,  12,  15,  18,  21,  24,  27,...
.  7,   7,   7,   7,   7,   7,   7,   7,   7,   7,...
.  0,   4,   8,  12,  16,  20,  24,  28,  32,  36,...
.  9,   9,   9,   9,   9,   9,   9,   9,   9,   9,...
.  0,   5,  10,  15,  20,  25,  30,  35,  40,  45,...
...

Rows: A000004, A000012, A001477, A010701, A005843, A010716, A008585, A010727, A008586, A010734, A008587.

Columns k: A026741 (k=1), A001477 (k=2), zero together with A080512 (k=3), A022998 (k=4), A195140 (k=5), zero together with A165998 (k=6), A195159 (k=7), A195161 (k=8), A195312 k=(9), A195817 (k=10), A317311 (k=11), A317312 (k=12), A317313 (k=13), A317314 k=(14), A317315 (k=15), A317316 (k=16), A317317 (k=17), A317318 (k=18), A317319 k=(19), A317320 (k=20), A317321 (k=21), A317322 (k=22), A317323 (k=23), A317324 k=(24), A317325 (k=25), A317326 (k=26).

T(n,k) = n*((k-2)*(-1)^n+k+2)/4 \\ Andrew Howroyd, Jul 23 2018

A195817 Multiples of 10 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 10, 3, 20, 5, 30, 7, 40, 9, 50, 11, 60, 13, 70, 15, 80, 17, 90, 19, 100, 21, 110, 23, 120, 25, 130, 27, 140, 29, 150, 31, 160, 33, 170, 35, 180, 37, 190, 39, 200, 41, 210, 43, 220, 45, 230, 47, 240, 49, 250, 51, 260, 53, 270, 55, 280, 57, 290, 59, 300

Offset: 0

Omar E. Pol, Sep 29 2011

A008592 and A005408 interleaved.
Partial sums give the generalized 14-gonal (or tetradecagonal) numbers A195818.
a(n) is also the length of the n-th line segment of a rectangular spiral on the infinite square grid. The vertices of the spiral are the generalized 14-gonal numbers. - Omar E. Pol, Jul 27 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Column 10 of A195151.
Sequences whose partial sums give the generalized n-gonal numbers, if n>=5: A026741, A001477, zero together with A080512, A022998, A195140, zero together with A165998, A195159, A195161, A195312, this sequence.
Cf. A091998, A195152.

[(2*(-1)^n+3)*n: n in [0..60]]; // Vincenzo Librandi, Sep 30 2011

With[{nn=30},Riffle[10*Range[0,nn],Range[1,2*nn+1,2]]] (* or *) LinearRecurrence[{0,2,0,-1},{0,1,10,3},70] (* Harvey P. Dale, Nov 24 2013 *)

a(n) = (2*(-1)^n+3)*n; \\ Andrew Howroyd, Jul 23 2018

a(n) = (2*(-1)^n+3)*n. - Vincenzo Librandi, Sep 30 2011
From Bruno Berselli, Sep 30 2011: (Start)
G.f.: x*(1+10*x+x^2)/((1-x)^2*(1+x)^2).
a(n) = -a(-n) = a(n-2)*n/(n-2) = 2*a(n-2)-a(n-4).
a(n) * a(n+1) = a(n(n+1)).
a(n) + a(n+1) = A091998(n+1). (End)
a(0)=0, a(1)=1, a(2)=10, a(3)=3, a(n)=2*a(n-2)-a(n-4). - Harvey P. Dale, Nov 24 2013
Multiplicative with a(2^e) = 5*2^e, a(p^e) = p^e for odd prime p. - Andrew Howroyd, Jul 23 2018
Dirichlet g.f.: zeta(s-1) * (1 + 2^(3-s)). - Amiram Eldar, Oct 25 2023

A211971 Column 0 of square array A211970 (in which column 1 is A000041).

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 16, 24, 36, 54, 78, 112, 160, 224, 312, 432, 590, 802, 1084, 1452, 1936, 2568, 3384, 4440, 5800, 7538, 9758, 12584, 16160, 20680, 26376, 33520, 42468, 53644, 67552, 84832, 106246, 132706, 165344, 205512, 254824, 315256, 389168, 479368

Offset: 0

Omar E. Pol, Jun 10 2012

nonn

Partial sums give A015128. - Omar E. Pol, Jan 09 2014

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A006950, A008794, A036820, A057077, A195152, A195848, A195849, A195850, A195851, A195852, A196933, A195825, A210964, A277643.

Flatten[{1, Differences[Table[Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}], {n, 0, 60}]]}] (* Vaclav Kotesovec, Oct 25 2016 *)
CoefficientList[Series[(1 - x)/EllipticTheta[4, 0, x], {x, 0, 43}], x] (* Robert G. Wilson v, Mar 06 2018 *)

a(n) ~ exp(Pi*sqrt(n))*Pi / (16*n^(3/2)) * (1 - (3/Pi + Pi/4)/sqrt(n) + (3/2 + 3/Pi^2+ Pi^2/24)/n). - Vaclav Kotesovec, Oct 25 2016, extended Nov 04 2016

G.f.: (1 - x)/theta_4(x), where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Mar 05 2018

A211970 Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 6, 3, 1, 1, 1, 10, 5, 2, 1, 1, 1, 16, 7, 3, 1, 1, 1, 1, 24, 11, 4, 2, 1, 1, 1, 1, 36, 15, 5, 3, 1, 1, 1, 1, 1, 54, 22, 7, 4, 2, 1, 1, 1, 1, 1, 78, 30, 10, 4, 3, 1, 1, 1, 1, 1, 1, 112, 42, 13, 5, 4, 2, 1, 1, 1, 1, 1, 1

Offset: 0

Omar E. Pol, Jun 10 2012

In the infinite square array if k is positive then column k is related to the generalized m-gonal numbers, where m = k+4. For example: column 1 is related to the generalized pentagonal numbers A001318. Column 2 is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on...
In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041. It seems unusual that the partition numbers are located in a middle column (see below row 1 of the table):
========================================================
.                                          Column k of
.                                          this square
.       Generalized   Triangle  Triangle   array A211970
k    m    m-gonal       "A"       "B"      [row sums of
.         numbers                          triangle "B"
.        (if k>=1)                         with a(0)=1,
.                                          if k >= 0]
========================================================
0    4    A008794        -         -         A211971
1    5    A001318     A195310   A175003      A000041
2    6    A000217     A195826   A195836      A006950
3    7    A085787     A195827   A195837      A036820
4    8    A001082     A195828   A195838      A195848
5    9    A118277     A195829   A195839      A195849
6   10    A074377     A195830   A195840      A195850
7   11    A195160     A195831   A195841      A195851
8   12    A195162     A195832   A195842      A195852
9   13    A195313     A195833   A195843      A196933
10  14    A195818     A210944   A210954      A210964
...
It appears that column 2 of the square array is A006950.
It appears that column 3 of the square array is A036820.
The partial sums of column 0 give A015128. - Omar E. Pol, Feb 09 2014

			Array begins:
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
2,     2,   1,   1,   1,   1,  1,  1,  1,  1,  1, ...
4,     3,   2,   1,   1,   1,  1,  1,  1,  1,  1, ...
6,     5,   3,   2,   1,   1,  1,  1,  1,  1,  1, ...
10,    7,   4,   3,   2,   1,  1,  1,  1,  1,  1, ...
16,   11,   5,   4,   3,   2,  1,  1,  1,  1,  1, ...
24,   15,   7,   4,   4,   3,  2,  1,  1,  1,  1, ...
36,   22,  10,   5,   4,   4,  3,  2,  1,  1,  1, ...
54,   30,  13,   7,   4,   4,  4,  3,  2,  1,  1, ...
78,   42,  16,  10,   5,   4,  4,  4,  3,  2,  1, ...
112,  56,  21,  12,   7,   4,  4,  4,  4,  3,  2, ...
160,  77,  28,  14,  10,   5,  4,  4,  4,  4,  3, ...
224, 101,  35,  16,  12,   7,  4,  4,  4,  4,  4, ...
312, 135,  43,  21,  13,  10,  5,  4,  4,  4,  4, ...
432, 176,  55,  27,  14,  12,  7,  4,  4,  4,  4, ...
...

L. Euler, De mirabilibus proprietatibus numerorum pentagonalium
L. Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem

Columns (0-10): A211971, A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
For another version see A195825.
Cf. A057077, A195152.

T(n,k) = A211971(n), if k = 0.

T(n,k) = A195825(n,k), if k >= 1.

A292551 Expansion of x(1 - 2x + x^2 + 7x^3 - x^4)/((1 - x)^4(1 + x)^3).

Original entry on oeis.org

0, 1, -1, 3, 4, 12, 21, 34, 56, 75, 115, 141, 204, 238, 329, 372, 496, 549, 711, 775, 980, 1056, 1309, 1398, 1704, 1807, 2171, 2289, 2716, 2850, 3345, 3496, 4064, 4233, 4879, 5067, 5796, 6004, 6821, 7050, 7960, 8211, 9219, 9493, 10604, 10902, 12121, 12444, 13776, 14125, 15575

Offset: 0

Ilya Gutkovskiy, Sep 18 2017

The n-th generalized n-gonal number (for n >= 5).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A006484, A060354, A100119, A195152.

Main diagonal of A303301.

List([0..50],n->(2*n^3-2*n^2+(-1)^n*(2*n^2-11*n-6)-5*n+6)/16); # Muniru A Asiru, Aug 08 2018

a:= n-> (m-> m*((n-2)*m-(n-4))/2)(-ceil(n/2)*(-1)^n):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 29 2018

CoefficientList[Series[x (1 - 2 x + x^2 + 7 x^3 - x^4)/((1 - x)^4 (1 + x)^3), {x, 0, 50}], x]
Table[SeriesCoefficient[x (1 + (n - 4) x + x^2)/((1 - x)^3 (1 + x)^2), {x, 0, n}], {n, 0, 50}]
LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {0, 1, -1, 3, 4, 12, 21}, 51]
Table[(2 n^3 - 2 n^2 + (-1)^n (2 n^2 - 11 n - 6) - 5 n + 6)/16, {n, 0, 50}]

x='x+O('x^99); concat(0, Vec(x*(1-2*x+x^2+7*x^3-x^4)/((1-x)^4*(1+x)^3))) \\ Altug Alkan, Sep 18 2017

G.f.: x*(1 - 2*x + x^2 + 7*x^3 - x^4)/((1 - x)^4*(1 + x)^3).
a(n) = [x^n] x*(1 + (n - 4)*x + x^2)/((1 - x)^3*(1 + x)^2).
E.g.f.: (1/16)*((-6 + 9*x + 2*x^2)*exp(-x) + (6 - 5*x + 4*x^2 + 2*x^3)*exp(x)).
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) = (2*n^3 - 2*n^2 + (-1)^n*(2*n^2 - 11*n - 6) - 5*n + 6)/16.

A303301 Square array T(n,k) read by antidiagonals upwards in which row n is obtained by taking the general formula for generalized n-gonal numbers: m((n - 2)m - n + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and n >= 5. Here n >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, -3, 0, 1, -2, 0, 0, 1, -1, 1, -8, 0, 1, 0, 2, -5, -3, 0, 1, 1, 3, -2, 0, -15, 0, 1, 2, 4, 1, 3, -9, -8, 0, 1, 3, 5, 4, 6, -3, -2, -24, 0, 1, 4, 6, 7, 9, 3, 4, -14, -15, 0, 1, 5, 7, 10, 12, 9, 10, -4, -5, -35, 0, 1, 6, 8, 13, 15, 15, 16, 6, 5, -20, -24, 0, 1, 7, 9, 16, 18, 21, 22, 16, 15, -5, -9, -48

Offset: 0

Omar E. Pol, Jun 08 2018

Note that the formula mentioned in the definition gives several kinds of numbers, for example:
Row 0 and row 1 give A317300 and A317301 respectively.
Row 2 gives A001057 (canonical enumeration of integers).
Row 3 gives 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Row 4 gives A008794 (squares repeated) except the initial zero.
Finally, for n >= 5 row n gives the generalized k-gonal numbers (see Crossrefs section).

			Array begins:
------------------------------------------------------------------
n\m  Seq. No.    0   1  -1   2  -2   3   -3    4   -4    5   -5
------------------------------------------------------------------
0    A317300:    0,  1, -3,  0, -8, -3, -15,  -8, -24, -15, -35...
1    A317301:    0,  1, -2,  1, -5,  0,  -9,  -2, -14,  -5, -20...
2    A001057:    0,  1, -1,  2, -2,  3,  -3,   4,  -4,   5,  -5...
3   (A008795):   0,  1,  0,  3,  1,  6,   3,  10,   6,  15,  10...
4   (A008794):   0,  1,  1,  4,  4,  9,   9,  16,  16,  25,  25...
5    A001318:    0,  1,  2,  5,  7, 12,  15,  22,  26,  35,  40...
6    A000217:    0,  1,  3,  6, 10, 15,  21,  28,  36,  45,  55...
7    A085787:    0,  1,  4,  7, 13, 18,  27,  34,  46,  55,  70...
8    A001082:    0,  1,  5,  8, 16, 21,  33,  40,  56,  65,  85...
9    A118277:    0,  1,  6,  9, 19, 24,  39,  46,  66,  75, 100...
10   A074377:    0,  1,  7, 10, 22, 27,  45,  52,  76,  85, 115...
11   A195160:    0,  1,  8, 11, 25, 30,  51,  58,  86,  95, 130...
12   A195162:    0,  1,  9, 12, 28, 33,  57,  64,  96, 105, 145...
13   A195313:    0,  1, 10, 13, 31, 36,  63,  70, 106, 115, 160...
14   A195818:    0,  1, 11, 14, 34, 39,  69,  76, 116, 125, 175...
15   A277082:    0,  1, 12, 15, 37, 42,  75,  82, 126, 135, 190...
...

Alois P. Heinz, Antidiagonals n = 0..200 (first 45 antidiagonals from Robert G. Wilson v)

Columns 0..2 are A000004, A000012, A023445.
Column 3 gives A001477 which coincides with the row numbers.
Main diagonal gives A292551.
Row 0-2 gives A317300, A317301, A001057.
Row 3 gives 0 together with A008795.
Row 4 gives A008794.
For n >= 5, rows n gives the generalized n-gonal numbers: A001318 (n=5), A000217 (n=6), A085787 (n=7), A001082 (n=8), A118277 (n=9), A074377 (n=10), A195160 (n=11), A195162 (n=12), A195313 (n=13), A195818 (n=14), A277082 (n=15), A274978 (n=16), A303305 (n=17), A274979 (n=18), A303813 (n=19), A218864 (n=20), A303298 (n=21), A303299 (n=22), A303303 (n=23), A303814 (n=24), A303304 (n=25), A316724 (n=26), A316725 (n=27), A303812 (n=28), A303815 (n=29), A316729 (n=30).
Cf. A194801, A195152.
Cf. A317302 (a similar table but with polygonal numbers).

t[n_, r_] := PolygonalNumber[n, If[OddQ@ r, Floor[(r + 1)/2], -r/2]]; Table[ t[n - r, r], {n, 0, 11}, {r, 0, n}] // Flatten (* also *)
(* to view the square array *)  Table[ t[n, r], {n, 0, 15}, {r, 0, 10}] // TableForm (* Robert G. Wilson v, Aug 08 2018 *)

T(n,k) = A194801(n-3,k) if n >= 3.

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

A235670 Square array read by antidiagonals upwards in which the n-th column gives the partial sums of the n-th column of A211970.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 4, 2, 1, 14, 7, 3, 2, 1, 24, 12, 5, 3, 2, 1, 40, 19, 8, 4, 3, 2, 1, 64, 30, 12, 6, 4, 3, 2, 1, 100, 45, 17, 9, 5, 4, 3, 2, 1, 154, 67, 24, 13, 7, 5, 4, 3, 2, 1, 232, 97, 34, 17, 10, 8, 6, 5, 4, 3, 2, 1, 344, 139, 47, 22, 14, 8, 6, 5, 4, 3, 2, 1

Offset: 0

Omar E. Pol, Jan 13 2014

The column 0 is related to A008794 in the same way as the column k is related to the generalized (k+4)-gonal numbers, for k >= 1. For more information see A195152 and A211970.

			Array begins:
1,     1,   1,   1,   1,   1,  1,  1,  1,  1,  1,...
2,     2,   2,   2,   2,   2,  2,  2,  2,  2,  2,...
4,     4,   3,   3,   3,   3,  3,  3,  3,  3,  3,...
8,     7,   5,   4,   4,   4,  4,  4,  4,  4,  4,...
14,   12,   8,   6,   5,   5,  5,  5,  5,  5,  5,...
24,   19,  12,   9,   7,   6,  6,  6,  6,  6,  6,...
40,   30,  17,  13,  10,   8,  7,  7,  7,  7,  7,...
64,   45,  24,  17,  14,  11,  9,  8,  8,  8,  8,...
100,  67,  34,  22,  18,  15, 12, 10,  9,  9,  9,...
154,  97,  47,  29,  22,  19, 16, 13, 11, 10, 10,...
232, 139,  63,  39,  27,  23, 20, 17, 14, 12, 11,...
344, 195,  84,  51,  34,  27, 24, 21, 18, 15, 13,...
504, 272, 112,  65,  44,  32, 28, 25, 22, 19, 16,...
728, 383, 147,  81,  56,  39, 32, 29, 26, 23, 20,...
...

Column 1 is A015128, the partial sums of A211971.
Column 2 is A000070, the partial sums of A000041.
Column 3 is A233969, the partial sums of A006950.
Cf. A000217, A001318, A008794, A085787, A057077, A195152, A211970.

T(n,k) = Sum_{j=0..n} A211970(j,k), (n>=0, k>=0).

cp's OEIS Frontend

A195825 Square array T(n,k) read by antidiagonals, n>=0, k>=1, which arises from a generalization of Euler's Pentagonal Number Theorem.

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A195151 Square array read by antidiagonals upwards: T(n,k) = n*((k-2)*(-1)^n+k+2)/4, n >= 0, k >= 0.

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A195817 Multiples of 10 and odd numbers interleaved.

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A211971 Column 0 of square array A211970 (in which column 1 is A000041).

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A211970 Square array read by antidiagonal: T(n,k), n >= 0, k >= 0, which arises from a generalization of Euler's Pentagonal Number Theorem.

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A292551 Expansion of x*(1 - 2*x + x^2 + 7*x^3 - x^4)/((1 - x)^4*(1 + x)^3).

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A303301 Square array T(n,k) read by antidiagonals upwards in which row n is obtained by taking the general formula for generalized n-gonal numbers: m*((n - 2)*m - n + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and n >= 5. Here n >= 0.

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

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A195151 Square array read by antidiagonals upwards: T(n,k) = n((k-2)(-1)^n+k+2)/4, n >= 0, k >= 0.

A292551 Expansion of x(1 - 2x + x^2 + 7x^3 - x^4)/((1 - x)^4(1 + x)^3).

A303301 Square array T(n,k) read by antidiagonals upwards in which row n is obtained by taking the general formula for generalized n-gonal numbers: m((n - 2)m - n + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and n >= 5. Here n >= 0.

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