A061315 -id:A061315 - cp's OEIS frontend

A061319 Column 4 of A061315.

0, 1, 10, 76, 430, 1870, 6601, 19810, 52326, 124785, 273790, 560626, 1083160, 1991626, 3509065, 5957260, 9789076, 15628185, 24317226, 36975520, 55067530, 80483326, 115632385, 163552126, 228032650, 313759225, 426474126, 573159510

Offset: 0

Henry Bottomley, Apr 24 2001

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

a(n) =A061318(n)/4 =sqrt(A061320(n)-A061318(n)).
G.f. -x*(1+x+22*x^2+22*x^3+22*x^4+x^5+x^6) / (x-1)^9. - R. J. Mathar, Aug 11 2012
a(n) = n*(n+1)*(n^2+n+4)*(n^4+2*n^3+5*n^2+4*n+36)/576 . R. J. Mathar, Aug 11 2012

A061321 Column 5 of A061315.

Original entry on oeis.org

0, 1, 28, 1216, 37324, 700876, 8719921, 78503068, 547643916, 3114359073, 14992411852, 62860750876, 234647983648, 793316418076

Offset: 0

Henry Bottomley, Apr 24 2001

nonn

a(n) =A061320(n)/5.

A006007 4-dimensional analog of centered polygonal numbers: a(n) = n(n+1)*(n^2+n+4)/12.

Original entry on oeis.org

0, 1, 5, 16, 40, 85, 161, 280, 456, 705, 1045, 1496, 2080, 2821, 3745, 4880, 6256, 7905, 9861, 12160, 14840, 17941, 21505, 25576, 30200, 35425, 41301, 47880, 55216, 63365, 72385, 82336, 93280, 105281, 118405, 132720, 148296, 165205, 183521

Offset: 0

N. J. A. Sloane, Simon Plouffe

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
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..710
Per Alexandersson, Sam Hopkins, and Gjergji Zaimi, Restricted Birkhoff polytopes and Ehrhart period collapse, arXiv:2206.02276 [math.CO], 2022.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
D.-N. Verma, Towards Classifying Finite Point-Set Configurations, 1997, Unpublished. [Scanned copy of annotated version of preprint given to me by the author in 1997. - _N. J. A. Sloane_, Oct 04 2021]
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A003215, A000537, A000578, A005898, A027602. - Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Cf. A000217, A061315, A061316, A061318.

[n*(n+1)*(n^2+n+4)/12: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

f[n_]:=n^3;lst={};s=0;Do[s+=(f[n]+f[n+1]+f[n+2]);AppendTo[lst,s/9],{n,0,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 03 2009 *)
Table[2Binomial[n+2,4]+Binomial[n+1,2],{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,5,16,40},40] (* Harvey P. Dale, Sep 30 2011 *)

a(n)=n*(n+1)*(n^2+n+4)/12 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: (1+x^2)/(1-x)^5.
a(n) = 2*binomial(n + 2, 4) + binomial(n + 1, 2).
a(n) = A061316(n)/3 = A061315(n, 3) = sqrt(A061318(n)-A061316(n)).
a(0)=0, a(1)=1, a(2)=5, a(3)=16, a(4)=40, a(n) = 5*a(n-1) -  10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Harvey P. Dale, Sep 30 2011
For n>0, a(n) = (A000217(n-1)^2 + A000217(n)^2 + A000217(n+1)^2 - 1)/9. - Richard R. Forberg, Dec 25 2013
Sum_{n>=1} 1/a(n) = 15/4 - tanh(sqrt(15)*Pi/2)*Pi*sqrt(3/5). - Amiram Eldar, Aug 23 2022
E.g.f.: exp(x)*(12 + 48*x + 42*x^2 + 12*x^3 + x^4)/12. - Stefano Spezia, Aug 31 2023

More terms from Henry Bottomley, Apr 24 2001

A061314 Table read by descending antidiagonals where T(n,k) = T(n,k-1) + T(n,k-1)^2/k^2 and T(n,0)=n.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 0, 3, 6, 3, 0, 4, 15, 12, 4, 0, 5, 40, 48, 20, 5, 0, 6, 140, 304, 120, 30, 6, 0, 7, 924, 6080, 1720, 255, 42, 7, 0, 8, 24640, 1484736, 186620, 7480, 483, 56, 8, 0, 9, 12415040, 61235956672, 1393267596, 3504380, 26404, 840, 72, 9, 0, 10

Offset: 0

Henry Bottomley, Apr 24 2001

Not always an integer.

			The table begins:
    0,       0,       0,       0,          0,...
    1,       2,       3,       4,          5,...
    2,       6,      15,      40,        140,...
    3,      12,      48,     304,       6080,...
    4,      20,     120,    1720,     186620,...
    5,      30,     255,    7480,    3504380,...
    6,      42,     483,   26404,   43599605,...
    7,      56,     840,   79240,  392515340,...
    8,      72,    1368,  209304, 2738219580,...
    ...

Rows include A000004, A000027 and A061322. Columns include A001477, A002378, A061316, A061318 and A061320.

T(n, k) = T(n, k-1) + A061315(n, k)^2.

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A061319 Column 4 of A061315.

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A061321 Column 5 of A061315.

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A006007 4-dimensional analog of centered polygonal numbers: a(n) = n(n+1)*(n^2+n+4)/12.

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A061314 Table read by descending antidiagonals where T(n,k) = T(n,k-1) + T(n,k-1)^2/k^2 and T(n,0)=n.

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