A004068 -id:A004068 - cp's OEIS frontend

A006414 Number of nonseparable toroidal tree-rooted maps with n + 2 edges and n + 1 vertices.

1, 9, 40, 125, 315, 686, 1344, 2430, 4125, 6655, 10296, 15379, 22295, 31500, 43520, 58956, 78489, 102885, 133000, 169785, 214291, 267674, 331200, 406250, 494325, 597051, 716184, 853615, 1011375, 1191640, 1396736, 1629144, 1891505, 2186625, 2517480, 2887221

Offset: 0

N. J. A. Sloane

The number of faces is 1.
a(n) = K(Oa(2,3,n)), Kekulé numbers of certain benzenoid structures (see the Cyvin - Gutman reference).
Sequence of partial sums of A006322. - L. Edson Jeffery, Dec 13 2011
The sequence b(n) = a(n-2) with a(-1) = 0, for n >= 1, is b(n) = n^3*(n^2 - 1)/4!. It is obtained by comparing the result for the powers n^5 from Worpitzky's identity (see a formula in A000584) with the result obtained from the counting of degrees of freedom for the decomposition of a rank 5 tensor in n dimensions via the standard Young tableaux version with 5 boxes corresponding to the seven partitions of 5. The difference of the two versions gives: 10*(binomial(n+3, 5) + 3*binomial(n+2, 5) + binomial(n+1, 5)) = 5*n*(binomial(n+2, 4) + binomial(n+1, 4)) = 10*b(n). See the formula for a(n) below. - Wolfdieter Lang, Jul 18 2019

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988, p. 105, eq. (ii), and p. 186.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B, Vol. 18, No. 3 (1975), pp. 222-259.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Differences of A006542 (C(n, 3)*C(n-1, 3)/4).

Cf. A004068, A005891, A006322, A006415, A006434, A006441, A133754, A143945.

List([0..40], n-> (n+2)^2*Binomial(n+3,3)/4 ); G. C. Greubel, Sep 02 2019

[(n+1)*(n+2)^3*(n+3)/24: n in [0..40]]; // Wesley Ivan Hurt, May 10 2014

seq((n+2)^2*binomial(n+3,3)/4, n=0..40); # G. C. Greubel, Sep 02 2019

Table[(n + 1)*(n + 2)^3*(n + 3)/24, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

a(n) = (n+1)*(n+2)^3*(n+3)/24; \\ Michel Marcus, Jul 09 2017

[(n+2)^2*binomial(n+3,3)/4 for n in (0..40)] # G. C. Greubel, Sep 02 2019

a(n) = (n+1)*(n+2)^3*(n+3)/24. - N. J. A. Sloane, Apr 02 2004
a(n) = (n+2)^3*((n+2)^2 - 1)/24. - Paul Richards, Mar 04 2007
G.f.: (1 + 3*x + x^2)/(1-x)^6. - Colin Barker, Feb 21 2012
a(n) = (Sum_{k=0..n+1} k*(n+1)*((n+1)^2 - k^2))/6 for n > 0, which is the sum of all areas of Pythagorean triangles with arms 2*k*(n+1) and (n+1)^2 - k^2 with hypotenuse k^2 + (n+1)^2. - J. M. Bergot, May 12 2014
a(n) = A143945(n+2)/8. - J. M. Bergot, Jun 14 2014
Sum_{n>=0} 1/a(n) = 30 - 24*zeta(3). - Jaume Oliver Lafont, Jul 09 2017
a(n) = binomial(n+5, 5) + 3*binomial(n+4, 5) + binomial(n+3, 5) = ((n+2)/2)*(binomial(n+4, 4) + binomial(n+3, 4)), for n >= 0. See a comment above on the sequence b(n) = a(n-2) = n^3*(n^2 - 1)/4!. - Wolfdieter Lang, Jul 19 2019
E.g.f.: (24 + 192*x + 276*x^2 + 124*x^3 + 20*x^4 + x^5)*exp(x)/4!. - G. C. Greubel, Sep 02 2019
Sum_{n>=0} (-1)^n/a(n) = 18*zeta(3) + 48*log(2) - 54. - Amiram Eldar, Jan 09 2022

More terms from Robert Newstedt (Patternfinder(AT)webtv.net)

Name clarified by Andrew Howroyd, Apr 05 2021

A215630 Triangle read by rows: T(n,k) = n^2 - n*k + k^2, 1 <= k <= n.

Original entry on oeis.org

1, 3, 4, 7, 7, 9, 13, 12, 13, 16, 21, 19, 19, 21, 25, 31, 28, 27, 28, 31, 36, 43, 39, 37, 37, 39, 43, 49, 57, 52, 49, 48, 49, 52, 57, 64, 73, 67, 63, 61, 61, 63, 67, 73, 81, 91, 84, 79, 76, 75, 76, 79, 84, 91, 100, 111, 103, 97, 93, 91, 91, 93, 97, 103, 111

Offset: 1

Reinhard Zumkeller, Nov 11 2012

T(n,k) = A093995(n,k) - A075362(n,k) + A133819(n,k) = 2*A070216(n,k) - A215631(n,k), 1 <= k <= n.

			The triangle begins:
.  1:     1
.  2:     3    4
.  3:     7    7    9
.  4:    13   12   13   16
.  5:    21   19   19   21   25
.  6:    31   28   27   28   31   36
.  7:    43   39   37   37   39   43   49
.  8:    57   52   49   48   49   52   57   64
.  9:    73   67   63   61   61   63   67   73   81
. 10:    91   84   79   76   75   76   79   84   91  100
. 11:   111  103   97   93   91   91   93   97  103  111  121
. 12:   133  124  117  112  109  108  109  112  117  124  133  144 .

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A004068 (row sums), A002061 (left edge), A000290 (right edge).

Cf. A003215 (central terms).

a215630 n k = a215630_tabl !! (n-1) !! (k-1)
a215630_row n = a215630_tabl !! (n-1)
a215630_tabl = zipWith3 (zipWith3 (\u v w -> u - v + w))
                        a093995_tabl a075362_tabl a133819_tabl

A322638 Numbers that are sums of consecutive centered pentagonal numbers (A005891).

Original entry on oeis.org

0, 1, 6, 7, 16, 22, 23, 31, 47, 51, 53, 54, 76, 82, 98, 104, 105, 106, 127, 141, 158, 174, 180, 181, 182, 226, 233, 247, 264, 276, 280, 286, 287, 322, 323, 331, 374, 391, 405, 407, 421, 427, 428, 456, 502, 504, 526, 548, 555, 586, 601, 602, 607, 608, 609, 654, 681, 683, 722

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Centered Pentagonal Number

Cf. A004068, A005891, A152043, A319184, A322610, A322611, A322640.

L:= [seq((5*n^2+5*n+2)/2,n=0..30)]: N:= L[-1]:
S:=[0,op(ListTools:-PartialSums(L))]:
R:=select(`<=`,{0,seq(seq(S[n]-S[m],m=1..n-1),n=1..nops(S))},N):
sort(convert(R,list)); # Robert Israel, Mar 19 2023

terms = 59;
nmax = 16; kmax = 9; (* empirical *)
T = Table[(5n^2 + 5n + 2)/2, {n, 0, nmax}];
Union[{0}, T, Table[k MovingAverage[T, k], {k, 2, kmax}] // Flatten][[1 ;; terms]] (* Jean-François Alcover, Dec 26 2018 *)

A008354 a(n) = (5n^2 + 1)n^2 / 6.

Original entry on oeis.org

0, 1, 14, 69, 216, 525, 1086, 2009, 3424, 5481, 8350, 12221, 17304, 23829, 32046, 42225, 54656, 69649, 87534, 108661, 133400, 162141, 195294, 233289, 276576, 325625, 380926, 442989, 512344, 589541, 675150, 769761, 873984, 988449, 1113806, 1250725, 1399896

Offset: 0

N. J. A. Sloane and J. H. Conway

Partial sums of A005902. - Jonathan Vos Post, Mar 14 2006

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A005901, A005902.

List([0..30], n -> (5*n^2+1)*n^2/6); # Muniru A Asiru, Feb 12 2018

a:= n-> 5*n^4/6 + n^2/6: seq(a(n), n=0..45);

Table[n^2 (5 n^2 + 1)/6, {n, 0, 30}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 14, 69, 216}, 30] (* Harvey P. Dale, Feb 12 2015 *)

From R. J. Mathar, Aug 10 2008: (Start)
O.g.f.: x*(1 + x)*(x^2 + 8*x + 1)/(1 - x)^5.
a(n) = n*A004068(n). (End)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4, a(0)=0, a(1)=1, a(2)=14, a(3)=69, a(4)=216. - Harvey P. Dale, Feb 12 2015

Definition corrected by R. J. Mathar, Aug 10 2008

A272039 a(n) = 10n^2 + 4n + 1.

Original entry on oeis.org

1, 15, 49, 103, 177, 271, 385, 519, 673, 847, 1041, 1255, 1489, 1743, 2017, 2311, 2625, 2959, 3313, 3687, 4081, 4495, 4929, 5383, 5857, 6351, 6865, 7399, 7953, 8527, 9121, 9735, 10369, 11023, 11697, 12391, 13105, 13839, 14593, 15367, 16161, 16975, 17809, 18663, 19537

Offset: 0

Vincenzo Librandi, Apr 20 2016

Polynomials from the table "Coefficients and roots of Ehrhart polynomials" in Beck et al. paper (see Links section):
. Cube:                   A000578;
. Cube minus corner:      A004068;
. Prism:                  A002411;
. Octahedron:             A005900;
. Square pyramid:         A000330;
. Bypyramid:              A006003;
. Unimodular tetrahedron: A000292;
. Fat tetrahedron:        A167875;
. Cyclic(2,5), which has the same polynomial form of this sequence.
a(n) for n = 0, -1, 1, -2, 2, -3, 3, ... gives all x such that (5*x - 3)/2 is a square.
Squares in sequence: 1, 49, 1385329, 101263969, 2880599856289, ...
Is this 1 followed by A228219?

M. Beck, J. A. De Loera, M. Develin, J. Pfeifle and R. P. Stanley, Coefficients and roots of Ehrhart Polynomials, page 19.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000292, A000330, A000578, A002411, A004068, A005900, A006003, A167875, A168668, A228219, A272124, A272130.

Magma
```
[10*n^2+4*n+1: n in [0..50]];
```

Table[10 n^2 + 4 n + 1, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{1,15,49},50] (* Harvey P. Dale, Dec 26 2021 *)

a(n)=10*n^2+4*n+1 \\ Charles R Greathouse IV, Jun 17 2017

O.g.f.: (1 + 12*x + 7*x^2)/(1 - x)^3.
E.g.f.: (1 + 14*x + 10*x^2)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*A168668(n) + 1.

Edited by Bruno Berselli, Apr 22 2016

A377976 Expansion of the 48th root of the series 2*E_2(x) - E_4(x), where E_2(x) and E_4(x) are the Eisenstein series of weight 2 and 4.

Original entry on oeis.org

1, -6, -894, -174420, -38431614, -9048710040, -2221653118116, -561444889080960, -144914324838755910, -38011797621225586602, -10098281618881696696392, -2710458654395655881518356, -733711171629600485187568404, -200033609249999737396399900920, -54867682197669353983111639906656

Offset: 0

Peter Bala, Nov 14 2024

Let R = 1 + x*Z[[x]] denote the set of integer power series with constant term equal to 1. Let P(n) = {g^n, g in R}. The Eisenstein series E_2(x) lies in P(4) and E_4(x) lies in P(8) (Heninger et al.).
We claim that the series 2*E_2(x) - E_4(x) belongs to P(48).
Proof.
E_2(x) = 1 - 24*Sum_{n >= 1} sigma_1(n)*x^n.
E_4(x) = 1 + 240*Sum_{n >= 1} sigma_3(n)*x^n.
Hence,
2*E_2(x) - E_4(x) = 1 - (288)*Sum_{n >= 1} ((1/6)*sigma_1(n) + (5/6)*sigma_3(n))*x^n belongs to the set R, since the polynomial (1/6)*k + (5/6)*k^3 has integer values for integer k. See A004068.
Hence, 2*E_2(x) - E_4(x) == 1 (mod 288) == 1 (mod (2^5)*(3^2)).
It follows from Heninger et al., Theorem 1, Corollary 2, that the series 2*E_2(x) - E_4(x) belongs to P((2^4)*3) = P(48). End Proof.
In a similar way we find that the series 3*E_2(x) - E_6(x) - 1 belongs to P(72) and the three series 3*E_4(x) - 2*E_6(x), 5*E_4(x) - 2*E_10(x) - 2 and 5*E_6(x) - 3*E_10(x) - 1 belong to P(288).

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A004068, A006352 (E_2), A004009 (E_4), A108091 ((E_4)^1/8), A289392 ((E_2)^(1/4)), A341871 - A341875, A377973, A377974, A377975, A377977.

with(numtheory):
E := proc (k) local n, t1; t1 := 1 - 2*k*add(sigma[k-1](n)*q^n, n = 1..30)/bernoulli(k); series(t1, q, 30) end:
seq(coeftayl((2*E(2) - E(4))^(1/48), q = 0, n),n = 0..20);

terms = 20; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}]; E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}]; CoefficientList[Series[(2*E2[x] - E4[x])^(1/48), {x, 0, terms}], x] (* Vaclav Kotesovec, Aug 03 2025 *)

a(n) ~ c * d^n / n^(49/48), where d = 295.8669385406700495308385233671383399895922733900742171390678012914822364544611... and c = -0.0205882497833853345146399243734199945444083043388859856935627869352251231763... - Vaclav Kotesovec, Aug 03 2025

A185909 Accumulation array of A185908, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 3, 7, 6, 4, 11, 14, 10, 5, 15, 23, 23, 15, 6, 19, 32, 38, 34, 21, 7, 23, 41, 54, 56, 47, 28, 8, 27, 50, 70, 80, 77, 62, 36, 9, 31, 59, 86, 105, 110, 101, 79, 45, 10, 35, 68, 102, 130, 145, 144, 128, 98, 55, 11, 39, 77, 118, 155, 181, 190, 182, 158, 119, 66, 12, 43, 86, 134, 180, 217, 238, 240, 224, 191, 142, 78, 13, 47, 95, 150, 205, 253, 287, 301, 295, 270, 227, 167, 91, 14, 51, 104, 166, 230, 289, 336, 364, 370, 355, 320, 266, 194, 105, 15, 55, 113, 182, 255

Offset: 1

Clark Kimberling, Feb 06 2011

A member of the accumulation chain ... < A185907 < A185908 < A185909 < ...

(See A144112 for definitions of weight array and accumulation array.)

			Northwest corner:
   1,  2,  3,  4,  5
   3,  7, 11, 15, 19
   6, 14, 23, 32, 41
  10, 23, 38, 54, 70

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A144112, A185907, A185908.
diag (1,7,...): A004068.
diag (2,11,...): A033994.
diag (3,14,...): A162147.

f[n_, 0] := 0; f[0, k_] := 0; (*needed for the weight array*)
f[n_, k_] := Min[n, k] + n - 1;
s[n_, k_] := Sum[f[i, j], {i, 1, n}, {j, 1, k}];
Table[s[n - k + 1, k], {n, 10}, {k, n, 1, -1}] // Flatten

A071910 a(n) = t(n)t(n+1)t(n+2), where t() are the triangular numbers.

Original entry on oeis.org

0, 18, 180, 900, 3150, 8820, 21168, 45360, 89100, 163350, 283140, 468468, 745290, 1146600, 1713600, 2496960, 3558168, 4970970, 6822900, 9216900, 12273030, 16130268, 20948400, 26910000, 34222500, 43120350, 53867268, 66758580, 82123650, 100328400, 121777920

Offset: 0

N. J. A. Sloane, Jun 13 2002

nonn

a(n) is also the number of three-dimensional cage assemblies such that the assembly is not a cube. See also A052149 for the two-dimensional version and to A059827 for the non-exclusive version. - Alejandro Rodriguez, Oct 20 2020

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A006542, (first differences of a(n) /18) A006414, (second differences of a(n) /18) A006322, (third differences of a(n) /18) A004068, (fourth differences of a(n) /18) A005891, (fifth differences of a(n) /18) A008706.

Join[{0},Times@@@Partition[Accumulate[Range[40]],3,1]] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,18,180,900,3150,8820,21168},40] (* Harvey P. Dale, Aug 08 2025 *)

t(n) = n*(n+1)/2;
a(n) = t(n)*t(n+1)*t(n+2); \\ Michel Marcus, Oct 21 2015

a(n) = 18*A006542(n+3). - Vladeta Jovovic, Jun 14 2002
G.f.: 18*x*(1+3*x+x^2)/(1-x)^7. - Vladeta Jovovic, Jun 14 2002
a(n) = ((n+1)*(n+2))^3/8 - Sum_{i=1..n+1} i^3. - Jon Perry, Feb 13 2004
a(n) = C(2+n, n)*C(3+n, 1+n)*C(4+n, 2+n). - Zerinvary Lajos, Jul 29 2005
a(n) = A059827(n+1) - A000537(n+1). - Michel Marcus, Oct 21 2015

A365904 Triangle read by rows T(n,k) = n^2 - binomial(k+1,2), n>=1, k

Original entry on oeis.org

1, 4, 3, 9, 8, 6, 16, 15, 13, 10, 25, 24, 22, 19, 15, 36, 35, 33, 30, 26, 21, 49, 48, 46, 43, 39, 34, 28, 64, 63, 61, 58, 54, 49, 43, 36, 81, 80, 78, 75, 71, 66, 60, 53, 45, 100, 99, 97, 94, 90, 85, 79, 72, 64, 55, 121, 120, 118, 115, 111, 106, 100, 93, 85, 76, 66

Offset: 1

Joan Llobera Querol, Sep 22 2023

T(n,k) is the number of points in a rhombus that has 1 point in the first row, then 2 in the second, and following until the n-th row with n points, and then n-1 in the following row, n-2 in the following to end with a row with k+1 points.
T(n,0) are the perfect squares (A000290).
T(n,n-1) are the triangular numbers (A000217).

			Triangle begins:
    1;
    4,  3;
    9,  8,  6;
   16, 15, 13, 10;
   25, 24, 22, 19, 15;
   36, 35, 33, 30, 26, 21;
   49, 48, 46, 43, 39, 34, 28;
   64, 63, 61, 58, 54, 49, 43, 36;
   81, 80, 78, 75, 71, 66, 60, 53, 45;
  100, 99, 97, 94, 90, 85, 79, 72, 64, 55;
  ...

Joan Llobera Querol, Table of n, a(n) for n = 1..10000
Zach Wissner-Gross, Can You Shape the Peloton?, Fiddler on the Proof, Sep 22, 2023.

Row sums give A004068.

Cf. A214859.

G.f.: x*(1 + x - 4*x^2*y + x^3*y^2 + x^4*y^2)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Oct 05 2023

A329523 a(n) = n * (binomial(n + 1, 3) + 1).

Original entry on oeis.org

0, 1, 4, 15, 44, 105, 216, 399, 680, 1089, 1660, 2431, 3444, 4745, 6384, 8415, 10896, 13889, 17460, 21679, 26620, 32361, 38984, 46575, 55224, 65025, 76076, 88479, 102340, 117769, 134880, 153791, 174624, 197505, 222564, 249935, 279756, 312169, 347320, 385359, 426440

Offset: 0

Ilya Gutkovskiy, Nov 15 2019

The n-th centered n-gonal pyramidal number.

			Square array begins:
  (0), 1,  2,   3,   4,    5,  ... A001477
   0, (1), 3,   7,  14,   25,  ... A004006
   0,  1, (4), 11,  24,   45,  ... A006527
   0,  1,  5, (15), 34,   65,  ... A006003 (partial sums of A005448)
   0,  1,  6,  19, (44),  85,  ... A005900 (partial sums of A001844)
   0,  1,  7,  23,  54, (105), ... A004068 (partial sums of A005891)
...
This sequence is the main diagonal of the array.

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), 142.

Kelvin Voskuijl, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000292, A000330, A002415, A002417, A006000, A006484, A008911, A050407, A060354, A100119, A188475 (first differences).

[ n*(Binomial(n+1,3)+1):n in [0..40]]; // Marius A. Burtea, Nov 15 2019

R:=PowerSeriesRing(Integers(), 41); [0] cat Coefficients(R!(x*(1-x+5*x^2-x^3)/(1-x)^5)); // Marius A. Burtea, Nov 15 2019

Table[n (Binomial[n + 1, 3] + 1), {n, 0, 40}]
nmax = 40; CoefficientList[Series[x (1 - x + 5 x^2 - x^3)/(1 - x)^5, {x, 0, nmax}], x]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 4, 15, 44}, 41]

G.f.: x * (1 - x + 5*x^2 - x^3) / (1 - x)^5.
E.g.f.: exp(x) * x * (1 + x + x^2 + x^3 / 6).
a(n) = n * (n + 2) * (n^2 - 2*n + 3) / 6.
a(n) = n * (A000292(n-1) + 1).
a(n) = n + 2 * Sum_{k=1..n} A000330(k-1).
a(n) + a(-n) = 4 * A002415(n).

cp's OEIS Frontend

A006414 Number of nonseparable toroidal tree-rooted maps with n + 2 edges and n + 1 vertices.

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A215630 Triangle read by rows: T(n,k) = n^2 - n*k + k^2, 1 <= k <= n.

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A322638 Numbers that are sums of consecutive centered pentagonal numbers (A005891).

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A008354 a(n) = (5*n^2 + 1)*n^2 / 6.

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A272039 a(n) = 10*n^2 + 4*n + 1.

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A377976 Expansion of the 48th root of the series 2*E_2(x) - E_4(x), where E_2(x) and E_4(x) are the Eisenstein series of weight 2 and 4.

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A185909 Accumulation array of A185908, by antidiagonals.

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Examples

A008354 a(n) = (5n^2 + 1)n^2 / 6.

A272039 a(n) = 10n^2 + 4n + 1.

A071910 a(n) = t(n)t(n+1)t(n+2), where t() are the triangular numbers.