A292551 -id:A292551 - cp's OEIS frontend

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.

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.

A301972 a(n) = n(n^2 - 2n + 4)binomial(2n,n)/((n + 1)*(n + 2)).

Original entry on oeis.org

0, 1, 4, 21, 112, 570, 2772, 13013, 59488, 266526, 1175720, 5123426, 22108704, 94645460, 402503220, 1702300725, 7165821120, 30043474230, 125523450360, 522857438070, 2172127120800, 9002522512620, 37233403401480, 153704429299746, 633442159732032, 2606543487445100, 10710790748646352, 43957192722175908

Offset: 0

Ilya Gutkovskiy, Mar 29 2018

nonn

For n > 2, a(n) is the n-th term of the main diagonal of iterated partial sums array of n-gonal numbers (in other words, a(n) is the n-th (n+2)-dimensional n-gonal number, see also example).

			For n = 5 we have:
----------------------------
0   1    2    3     4    [5]
----------------------------
0,  1,   5,  12,   22,   35,  ... A000326 (pentagonal numbers)
0,  1,   6,  18,   40,   75,  ... A002411 (pentagonal pyramidal numbers)
0,  1,   7,  25,   65,  140,  ... A001296 (4-dimensional pyramidal numbers)
0,  1,   8,  33,   98,  238,  ... A051836 (partial sums of A001296)
0,  1,   9,  42,  140,  378,  ... A051923 (partial sums of A051836)
0,  1,  10,  52,  192, [570], ... A050494 (partial sums of A051923)
----------------------------
therefore a(5) = 570.

Index to sequences related to polygonal numbers

Cf. A000984, A002457, A006484, A057145, A060354, A080851, A080852, A100119, A180266, A275490, A292551.

Table[n (n^2 - 2 n + 4) Binomial[2 n, n]/((n + 1) (n + 2)), {n, 0, 27}]
nmax = 27; CoefficientList[Series[(-4 + 31 x - 66 x^2 + 28 x^3 + (4 - 7 x) (1 - 4 x)^(3/2))/(2 x^2 (1 - 4 x)^(3/2)), {x, 0, nmax}], x]
nmax = 27; CoefficientList[Series[Exp[2 x] (4 - x + 2 x^2) BesselI[1, 2 x]/x - 2 Exp[2 x] (2 - x) BesselI[0, 2 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[SeriesCoefficient[x (1 - 3 x + n x)/(1 - x)^(n + 3), {x, 0, n}], {n, 0, 27}]

O.g.f.: (-4 + 31*x - 66*x^2 + 28*x^3 + (4 - 7*x)*(1 - 4*x)^(3/2))/(2*x^2*(1 - 4*x)^(3/2)).
E.g.f.: exp(2*x)*(4 - x + 2*x^2)*BesselI(1,2*x)/x - 2*exp(2*x)*(2 - x)*BesselI(0,2*x).
a(n) = [x^n] x*(1 - 3*x + n*x)/(1 - x)^(n+3).
a(n) ~ 4^n*sqrt(n)/sqrt(Pi).
D-finite with recurrence: -(n+2)*(961*n-3215)*a(n) +4*(2081*n^2-4414*n-4668)*a(n-1) -28*(320*n-389)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 27 2020

A301973 a(n) = (n^2 - 3n + 6)binomial(n+2,3)/4.

Original entry on oeis.org

0, 1, 4, 15, 50, 140, 336, 714, 1380, 2475, 4180, 6721, 10374, 15470, 22400, 31620, 43656, 59109, 78660, 103075, 133210, 170016, 214544, 267950, 331500, 406575, 494676, 597429, 716590, 854050, 1011840, 1192136, 1397264, 1629705, 1892100, 2187255, 2518146, 2887924, 3299920, 3757650, 4264820

Offset: 0

Ilya Gutkovskiy, Mar 29 2018

For n > 2, a(n) is the n-th term of the partial sums of n-gonal pyramidal numbers (in other words, a(n) is the n-th 4-dimensional n-gonal number).

Index to sequences related to pyramidal numbers
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A000292, A000332, A001296, A002415, A006484, A060354, A080852, A256859, A291288 (partial sums), A292551.

Table[(n^2 - 3 n + 6) Binomial[n + 2, 3]/4, {n, 0, 40}]
nmax = 40; CoefficientList[Series[x (1 - 2 x + 6 x^2)/(1 - x)^6, {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[x] x (24 + 24 x + 24 x^2 + 10 x^3 + x^4)/24, {x, 0, nmax}], x] Range[0, nmax]!
Table[SeriesCoefficient[x (1 - 3 x + n x)/(1 - x)^5, {x, 0, n}], {n, 0, 40}]
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 4, 15, 50, 140}, 41]

O.g.f.: x*(1 - 2*x + 6*x^2)/(1 - x)^6.
E.g.f.: exp(x)*x*(24 + 24*x + 24*x^2 + 10*x^3 + x^4)/24.
a(n) = [x^n] x*(1 - 3*x + n*x)/(1 - x)^5.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

A341768 a(n) = n * (binomial(n,2) - 2).

Original entry on oeis.org

0, -2, -2, 3, 16, 40, 78, 133, 208, 306, 430, 583, 768, 988, 1246, 1545, 1888, 2278, 2718, 3211, 3760, 4368, 5038, 5773, 6576, 7450, 8398, 9423, 10528, 11716, 12990, 14353, 15808, 17358, 19006, 20755, 22608, 24568, 26638, 28821, 31120, 33538, 36078, 38743, 41536, 44460

Offset: 0

Ilya Gutkovskiy, Feb 19 2021

The n-th second n-gonal number.

			a(7) = A147875(7) = A000566(-7) = 133.

Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005449, A005564, A006002, A014105, A033954, A034856, A045944, A060354, A062728, A135705, A147875, A179986, A292551.

Table[n (Binomial[n, 2] - 2), {n, 0, 45}]
LinearRecurrence[{4, -6, 4, -1}, {0, -2, -2, 3}, 46]
CoefficientList[Series[-x (2 - 6 x + x^2)/(1 - x)^4, {x, 0, 45}], x]

G.f.: -x*(2 - 6*x + x^2)/(1 - x)^4.
E.g.f.: -exp(x)*x*(4 - 2*x - x^2)/2.
a(n) = n^2*(n - 1)/2 - 2*n.

cp's OEIS Frontend

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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A301972 a(n) = n*(n^2 - 2*n + 4)*binomial(2*n,n)/((n + 1)*(n + 2)).

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A301973 a(n) = (n^2 - 3*n + 6)*binomial(n+2,3)/4.

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A341768 a(n) = n * (binomial(n,2) - 2).

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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.

A301972 a(n) = n(n^2 - 2n + 4)binomial(2n,n)/((n + 1)*(n + 2)).

A301973 a(n) = (n^2 - 3n + 6)binomial(n+2,3)/4.