A002721 -id:A002721 - cp's OEIS frontend

A244879 Number of magic labelings of the cycle-of-loops graph LOOP X C_6 having magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph.

1, 18, 129, 571, 1884, 5103, 11998, 25362, 49347, 89848, 154935, 255333, 404950, 621453, 926892, 1348372, 1918773, 2677518, 3671389, 4955391, 6593664, 8660443, 11241066, 14433030, 18347095, 23108436, 28857843, 35752969, 43969626, 53703129, 65169688, 78607848

Offset: 0

N. J. A. Sloane, Jul 08 2014

Colin Barker, Table of n, a(n) for n = 0..1000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A019298, A006325, A244497, A292281, A244873, A244880.

CoefficientList[Series[(1 + 11 x + 24 x^2 + 11 x^3 + x^4)/(1 - x)^7, {x, 0, 31}], x] (* Michael De Vlieger, Sep 15 2017 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,18,129,571,1884,5103,11998},40] (* Harvey P. Dale, Jul 30 2019 *)

Vec((1 + 11*x + 24*x^2 + 11*x^3 + x^4) / (1 - x)^7 + O(x^40)) \\ Colin Barker, Jan 11 2017

G.f.: (1 + 11*x + 24*x^2 + 11*x^3 + x^4) / (1 - x)^7.
From Colin Barker, Jan 11 2017: (Start)
a(n) = (120 + 438*n + 677*n^2 + 570*n^3 + 275*n^4 + 72*n^5 + 8*n^6) / 120.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n > 6.
(End)

Name corrected by David J. Seal, Sep 13 2017

A061927 a(n) = n(n+1)(2n+1)(n^2+n+3)/30.

Original entry on oeis.org

0, 1, 9, 42, 138, 363, 819, 1652, 3060, 5301, 8701, 13662, 20670, 30303, 43239, 60264, 82280, 110313, 145521, 189202, 242802, 307923, 386331, 479964, 590940, 721565, 874341, 1051974, 1257382, 1493703, 1764303, 2072784, 2422992, 2819025

Offset: 0

Henry Bottomley, May 17 2001

Also number of magic labelings of the cubical graph of magic sum n-1 [Ahmed]. - R. J. Mathar, Jan 25 2007
If Y_i (i=1,2,3) are 2-blocks of a (n+3)-set X then a(n-4) is the number of 8-subsets of X intersecting each Y_i (i=1,2,3). - Milan Janjic, Oct 28 2007
The cube graph is also the prism graph I X C_4, so this is related to the number of magic labelings of other prism & related graphs. - David J. Seal, Sep 13 2017

Harry J. Smith, Table of n, a(n) for n = 0..1000
Maya Mohsin Ahmed, Algebraic Combinatorics of Magic Squares, arXiv:math/0405476 [math.CO], 2004, p. 73.
Shalosh B. Ekhad and Doron Zeilberger, There are (1/30)(r+1)(r+2)(2r+3)(r^2+3r+5) Ways For the Four Teams of a World Cup Group to Each Have r Goals For and r Goals Against [Thanks to the Soccer Analog of Prop. 4.6.19 of Richard Stanley's (Classic!) EC1], arXiv:1407.1919 [math.CO], 2014.
David Galvin and Courtney Sharpe, Independent set sequence of linear hyperpaths, arXiv:2409.15555 [math.CO], 2024. See p. 7.
Yu-hong Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq. (5), m=3.
Milan Janjic, Two Enumerative Functions
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Milan Janjic and Boris Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973. [Cached copy, with permission] See p. 32.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A006325, A019298, A244497, A244873, A289992, A292281, partial sums of A014820, A006975 (binomial transform shifted left).

Table[n (n + 1) (2 n + 1) (n^2 + n + 3)/30, {n, 0, 33}] (* or *)
CoefficientList[Series[x (1 + x)^3/(-1 + x)^6, {x, 0, 33}], x] (* Michael De Vlieger, Sep 15 2017 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,1,9,42,138,363},40] (* Harvey P. Dale, Apr 18 2018 *)

a(n) = { n*(n + 1)*(2*n + 1)*(n^2 + n + 3)/30 } \\ Harry J. Smith, Jul 29 2009

a(n) = a(n-1) + A014820(n) = A061926(9, n).

G.f.: x*(1+x)^3/(-1+x)^6 = 20/(-1+x)^5 + 1/(-1+x)^2 + 7/(-1+x)^3 + 18/(-1+x)^4 + 8/(-1+x)^6. - R. J. Mathar, Nov 18 2007

A244880 Number of magic labelings of the cycle-of-loops graph LOOP X C_8 having magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph.

Original entry on oeis.org

1, 47, 650, 4726, 23219, 87677, 274132, 743724, 1806597, 4016683, 8306078, 16168802, 29904823, 52936313, 90209192, 148694104, 238002057, 371131047, 565361074, 843316046, 1234212155, 1775313397, 2513615996, 3507784580, 4830364045, 6570292131, 8835738822, 11757299770, 15491572031

Offset: 0

N. J. A. Sloane, Jul 08 2014

Colin Barker, Table of n, a(n) for n = 0..1000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A019298, A006325, A244497, A244879, A244873, A289992.

A244880:=n->(630 + 3051*n + 6570*n^2 + 8211*n^3 + 6503*n^4 + 3339*n^5 + 1085*n^6 + 204*n^7 + 17*n^8) / 630: seq(A244880(n), n=0..50); # Wesley Ivan Hurt, Sep 16 2017

CoefficientList[Series[(1 + 38 (x + x^5) + 263 (x^2 + x^4) + 484 x^3 + x^6)/(1 - x)^9, {x, 0, 28}], x] (* Michael De Vlieger, Sep 15 2017 *)

Vec((1 + 6*x + x^2)*(1 + 32*x + 70*x^2 + 32*x^3 + x^4) / (1 - x)^9 + O(x^30)) \\ Colin Barker, Jan 12 2017

G.f.: (1+38*(x+x^5)+263*(x^2+x^4)+484*x^3+x^6) / (1-x)^9.
From Colin Barker, Jan 12 2017: (Start)
a(n) = (630 + 3051*n + 6570*n^2 + 8211*n^3 + 6503*n^4 + 3339*n^5 + 1085*n^6 + 204*n^7 + 17*n^8) / 630.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>8.
(End)
(326*n^2-195*n+142)*a(n) +(-652*n^2-652*n-10725)*a(n-1) +(326*n^2+847*n+663)*a(n-2) +2*(-165*n^2-165*n-71)=0. - R. J. Mathar, Mar 10 2025

Name corrected by David J. Seal, Sep 13 2017

A085461 Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1 <= v5, v2 <= v5, v2 <= v4 and v3 <= v4.

Original entry on oeis.org

1, 13, 70, 246, 671, 1547, 3164, 5916, 10317, 17017, 26818, 40690, 59787, 85463, 119288, 163064, 218841, 288933, 375934, 482734, 612535, 768867, 955604, 1176980, 1437605, 1742481, 2097018, 2507050, 2978851, 3519151, 4135152, 4834544

Offset: 1

Goran Kilibarda, Vladeta Jovovic, Jul 01 2003

Number of monotone n-weightings of a certain connected bipartite digraph. A monotone n-(vertex) weighting of a digraph D=(V,E) is a function w: V -> {0,1,...,n-1} such that w(v1) <= w(v2) for every arc (v1,v2) from E.
Kekulé numbers for certain benzenoids. - Emeric Deutsch, Nov 18 2005
Can be constructed by taking the product of the three members of a Pythagorean triples and dividing by 60. Formula: n*(n^2-1)*(n^2+1)/240 where n runs through the odd numbers >= 3. - Pierre Gayet, Apr 04 2009
Number of composable morphisms in a height-n tower of retractions. A retraction between objects X and Y is a pair of maps s:X->Y and r:Y->X such that r(s(x))=x for all x in X. Given objects X_0,X_1,X_2,...,X_n, we can ask for retractions s_i:X_i->X_{i+1},r_i:X_{i+1}->X_i, for each 0 <= i < n. The total number of morphisms in that category is 0^2 + 1^2 + 2^2 + ... + n^2 (cf. A000330). The total number of composable pairs of morphisms in that category is the sequence given here. - David Spivak, Feb 26 2014

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 168).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Daeseok Lee and H.-K. Ju, An Extension of Hibi's palindromic theorem, arXiv preprint arXiv:1503.05658 [math.CO], 2015.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
See p. 31
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A006322, A006325, A079547, A085462, A085463, A085464, A085465.

Rest[CoefficientList[Series[x*(1 + x)*(1 + 6*x + x^2)/(1 - x)^6, {x, 0, 50}], x]] (* G. C. Greubel, Oct 06 2017 *)

x='x+O('x^50); Vec(x*(1+x)*(1+6*x+x^2)/(1-x)^6) \\ G. C. Greubel, Oct 06 2017

a(n) = n + 11*binomial(n, 2) + 34*binomial(n, 3) + 40*binomial(n, 4) + 16*binomial(n, 5) = 1/30*n*(n+1)*(2*n+1)*(2*n^2 + 2*n + 1).
From Bruno Berselli, Dec 27 2010: (Start)
G.f.: x*(1+x)*(1+6*x+x^2)/(1-x)^6.
a(n) = ( n*A110450(n) - Sum_{i=0..n-1} A110450(i) )/3. (End)

A244869 Number of magic labelings with magic sum n of first graph shown in link.

Original entry on oeis.org

1, 9, 43, 143, 379, 859, 1738, 3226, 5597, 9197, 14453, 21881, 32095, 45815, 63876, 87236, 116985, 154353, 200719, 257619, 326755, 410003, 509422, 627262, 765973, 928213, 1116857, 1335005, 1585991, 1873391, 2201032, 2573000, 2993649, 3467609, 3999795, 4595415, 5259979, 5999307, 6819538

Offset: 0

N. J. A. Sloane, Jul 08 2014

Colin Barker, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Graphs for A244869-A244876.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1).

Cf. A244869-A244876.

LinearRecurrence[{5,-9,5,5,-9,5,-1},{1,9,43,143,379,859,1738},50] (* Paolo Xausa, Dec 06 2023 *)

Vec((1+4*x+7*x^2+4*x^3+x^4) / ((1-x)^6*(1+x)) + O(x^40)) \\ Colin Barker, Jan 11 2017

G.f.: (1 + 4*x + 7*x^2 + 4*x^3 + x^4) / ((1 - x)^6*(1 + x)).
From Colin Barker, Jan 11 2017: (Start)
a(n) = (15*(63 + (-1)^n) + 2592*n + 2880*n^2 + 1660*n^3 + 510*n^4 + 68*n^5) / 960.
a(n) = 5*a(n-1) - 9*a(n-2) + 5*a(n-3) + 5*a(n-4) - 9*a(n-5) + 5*a(n-6) - a(n-7) for n>6.
(End)

A244876 Number of magic labelings with magic sum n of 8th graph shown in link.

Original entry on oeis.org

1, 18, 154, 813, 3157, 9880, 26429, 62713, 135470, 271285, 510485, 911840, 1558368, 2564093, 4082142, 6313934, 9519951, 14031732, 20265700, 28738335, 40083439, 55070862, 74627587, 99860383, 132081092, 172833583, 223923623, 287450506, 365841890, 461890475, 578794188

Offset: 0

N. J. A. Sloane, Jul 08 2014

N. J. A. Sloane, Graphs for A244869-A244876.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (5, -7, -5, 22, -14, -14, 22, -5, -7, 5, -1).

Cf. A244869-A244876.

Table[3 n^7/160 + 63 n^6/320 + 151 n^5/160 + 339 n^4/128 + 2251 n^3/480 + n^2 (-1)^n/64 + 1677 n^2/320 + 3 n (-1)^n/64 + 3259 n/960 + 9 (-1)^n/256 + 247/256, {n,0,30}] (* Bruno Berselli, Jul 08 2014 *)

G.f.: (1+13*x+71*x^2+174*x^3+238*x^4+174*x^5+71*x^6+13*x^7+x^8)/((1-x)^8*(1+x)^3).

a(n) = 3*n^7/160 + 63*n^6/320 + 151*n^5/160 + 339*n^4/128 + 2251*n^3/480 + n^2*(-1)^n/64 + 1677*n^2/320 + 3*n*(-1)^n/64 + 3259*n/960 + 9*(-1)^n/256 + 247/256. [Bruno Berselli, Jul 08 2014]

A289992 Number of magic labelings of the prism graph I X C_8 having magic sum n.

Original entry on oeis.org

1, 49, 746, 6122, 34067, 144963, 506772, 1524628, 4074949, 9898229, 22220990, 46695870, 92769495, 175610631, 318756136, 557659432, 944355593, 1553488697, 2489980818, 3898657938, 5976186139, 8985711691, 13274641084, 19296041660, 27634190285

Offset: 0

David J. Seal, Sep 13 2017

R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973. [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A019298, A061927, A244497, A292281, A244873, A244880.

a(n) = A244880(n) + 2*Sum_{i=0..n-1} A244880(i).
From Colin Barker, Sep 13 2017: (Start)
G.f.: (1 + x)*(1 + 6*x + x^2)*(1 + 32*x + 70*x^2 + 32*x^3 + x^4) / (1 - x)^10.
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n>9. (End)
[Proof of the g.f. follows from the convolution formula and insertion of the g.f. A244880(x): Sum_{n>=0} a(n)x^n = Sum_{n>=0} A244880(n)*x^n +2*Sum_{n>=0} Sum_{i=0..n-1} A244880(i)*x^n = A244880(x) +2*Sum_{i>=0} Sum_{n>=i+1} A244880(i)*x^n = A244880(x) +2*Sum_{i>=0} A244880(i)*x^(i+1) Sum_{n>=0} x^n = A244880(x)+2*A244880(x)*x/(1+x) = A244880(x)*(1+2*x/(1-x)). R. J. Mathar, Mar 09 2025]

A292281 Number of magic labelings of the prism graph I X C_6 having magic sum n.

Original entry on oeis.org

1, 20, 167, 867, 3322, 10309, 27410, 64770, 139479, 278674, 523457, 933725, 1594008, 2620411, 4168756, 6444020, 9711165, 14307456, 20656363, 29283143, 40832198, 56086305, 75987814, 101661910, 134442035, 175897566, 227863845, 292474657, 372197252, 469870007, 588742824

Offset: 0

David J. Seal, Sep 13 2017

R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973. [Cached copy, with permission]
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A019298, A061927, A244497, A244879, A244873, A289992.

f[n_] := SeriesCoefficient[(1 + 11 x + 24 x^2 + 11 x^3 + x^4)/(1 - x)^7, {x, 0, n}]; Table[f[n] + 2 Sum[f[i], {i, 0, n - 1}], {n, 0, 24}] (* Michael De Vlieger, Sep 15 2017 *)

a(n) = A244879(n) + 2*Sum_{i=0..n-1} A244879(i).
From Colin Barker, Sep 13 2017: (Start)
G.f.: (1 + x)*(1 + 11*x + 24*x^2 + 11*x^3 + x^4) / (1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>7.
(End)
[Proof of the g.f. follows from the g.f. of A244879 with the resummation demonstrated in A289992: g.f. = A244879(x)*(1+2*x/(1-x)). - R. J. Mathar, Mar 09 2025]

A053494 Number of symmetric 5 X 5 matrices of nonnegative integers with every row and column adding to n.

Original entry on oeis.org

1, 26, 348, 2698, 14751, 62781, 222190, 681460, 1865715, 4655535, 10756921, 23290026, 47700173, 93104473, 174248451, 314246511, 548380980, 929209095, 1533389605, 2470568045, 3894914166, 6019752376, 9136114923, 13635769173, 20039850376, 29033765566

Offset: 0

N. J. A. Sloane, Jan 15 2000

R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Prop. 4.6.21, p. 235, G_5(lambda).

Colin Barker, Table of n, a(n) for n = 0..1000
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
R. P. Stanley, Magic labelings of graphs, symmetric magic squares,..., Duke Math. J. 43 (3) (1976) 511-531, F_5(x) in Section 5.
Index entries for linear recurrences with constant coefficients, signature (5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1).

Cf. A019298, A053493.

CoefficientList[Series[(1+21x+222x^2+1082x^3+3133x^4+5722x^5+7013x^6+5722x^7+3133x^8+1082x^9+222x^10+21x^11+x^12)/((1-x)^11(1+x)^6),{x,0,30}],x] (* or *) LinearRecurrence[ {5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1},{1,26,348,2698,14751,62781,222190,681460,1865715,4655535,10756921,23290026,47700173,93104473,174248451,314246511,548380980},30] (* Harvey P. Dale, Mar 05 2023 *)

Vec((1 + 21*x + 222*x^2 + 1082*x^3 + 3133*x^4 + 5722*x^5 + 7013*x^6 + 5722*x^7 + 3133*x^8 + 1082*x^9 + 222*x^10 + 21*x^11 + x^12) / ((1 - x)^11*(1 + x)^6) + O(x^30)) \\ Colin Barker, Jan 14 2017

G.f.: (1 + 21*x + 222*x^2 + 1082*x^3 + 3133*x^4 + 5722*x^5 + 7013*x^6 + 5722*x^7 + 3133*x^8 + 1082*x^9 + 222*x^10 + 21*x^11 + x^12) / ((1-x)^11*(1+x)^6).

a(n) = (189*(59981+5555*(-1)^n) + 18*(2345165+65331*(-1)^n)*n + (76615494+689850*(-1)^n)*n^2 + 40*(2138179+6237*(-1)^n)*n^3 + (63277966+47250*(-1)^n)*n^4 + 1260*(25421+3*(-1)^n)*n^5 + 11171664*n^6 + 2644080*n^7 + 405954*n^8 + 36500*n^9 + 1460*n^10) / 12386304. - Colin Barker, Jan 14 2017

Revised definition, Jul 06 2014

A244881 Expansion of (1 + 26x + 109x^2 + 109x^3 + 26x^4 + x^5) / (1 - x)^8.

Original entry on oeis.org

1, 34, 353, 2037, 8272, 26585, 72302, 173502, 377739, 760804, 1437799, 2576795, 4415346, 7280131, 11609996, 17982668, 27145413, 40049910, 57891613, 82153873, 114657092, 157613181, 213685594, 286055210, 378492335, 495435096, 642074499, 824446423, 1049530822

Offset: 0

N. J. A. Sloane, Jul 08 2014

Colin Barker, Table of n, a(n) for n = 0..1000
Rida Ait El Manssour and Anna-Laura Sattelberger, Combinatorial Differential Algebra of x^p, arXiv:2102.03182 [math.AG], 2021. Mentions this sequence.
R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]. See page 31.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Vec((1 + x)*(1 + 25*x + 84*x^2 + 25*x^3 + x^4) / (1 - x)^8 + O(x^30)) \\ Colin Barker, Jan 12 2017

From Colin Barker, Jan 12 2017: (Start)
a(n) = (2520 + 11526*n + 22617*n^2 + 24724*n^3 + 16275*n^4 + 6454*n^5 + 1428*n^6 + 136*n^7) / 2520.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>7.
(End)

cp's OEIS Frontend

A244879 Number of magic labelings of the cycle-of-loops graph LOOP X C_6 having magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph.

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A061927 a(n) = n*(n+1)*(2*n+1)*(n^2+n+3)/30.

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A244880 Number of magic labelings of the cycle-of-loops graph LOOP X C_8 having magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph.

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A085461 Number of 5-tuples (v1,v2,v3,v4,v5) of nonnegative integers less than n such that v1 <= v5, v2 <= v5, v2 <= v4 and v3 <= v4.

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A244869 Number of magic labelings with magic sum n of first graph shown in link.

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A244876 Number of magic labelings with magic sum n of 8th graph shown in link.

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A289992 Number of magic labelings of the prism graph I X C_8 having magic sum n.

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A292281 Number of magic labelings of the prism graph I X C_6 having magic sum n.

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A061927 a(n) = n(n+1)(2n+1)(n^2+n+3)/30.

A244881 Expansion of (1 + 26x + 109x^2 + 109x^3 + 26x^4 + x^5) / (1 - x)^8.