A060354 -id:A060354 - cp's OEIS frontend

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

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).

A366203 a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(3n,n-k-1) (n-3)^k.

Original entry on oeis.org

1, 2, 12, 156, 3507, 115692, 5066364, 276943568, 18152243967, 1387267590540, 121106707350928, 11889022355301672, 1296359140925188212, 155440199716271334648, 20327081449263918542412, 2879054747404226046119448, 439060192463001381367975215, 71727764882350305085962745740

Offset: 1

Ilya Gutkovskiy, Oct 04 2023

nonn

a(n) is the coefficient of x^n in expansion of series reversion of g.f. for n-gonal numbers (with signs).

Cf. A001764, A060354, A113207, A179848, A263843, A323208, A365816, A365817, A365818, A366204.

Unprotect[Power]; 0^0 = 1; Table[1/n Sum[Binomial[n + k - 1, k] Binomial[3 n, n - k - 1] (n - 3)^k, {k, 0, n - 1}], {n, 1, 18}]
Table[Binomial[3 n, n - 1] Hypergeometric2F1[1 - n, n, 2 (n + 1), 3 - n]/n, {n, 1, 18}]
Table[SeriesCoefficient[InverseSeries[Series[x (1 - (n - 3) x)/(1 + x)^3, {x, 0, n}], x], {x, 0, n}], {n, 1, 18}]

a(n) = [x^n] Series_Reversion( x * (1 - (n - 3) * x) / (1 + x)^3 ).

A117665 n times the n-th n-gonal number.

Original entry on oeis.org

0, 2, 6, 24, 80, 210, 462, 896, 1584, 2610, 4070, 6072, 8736, 12194, 16590, 22080, 28832, 37026, 46854, 58520, 72240, 88242, 106766, 128064, 152400, 180050, 211302, 246456, 285824, 329730, 378510, 432512, 492096, 557634, 629510, 708120, 793872

Offset: 0

Luc Stevens (lms022(AT)yahoo.com), Apr 11 2006

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A060354.

LinearRecurrence[{5,-10,10,-5,1},{0,2,6,24,80},50] (* Harvey P. Dale, Jan 02 2020 *)

a(n)=n*(n+1)*(n^2-3*n+4)/2 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = n (n + 1) (n^2 - 3 n + 4)/2. - corrected by Eric Rowland, Aug 15 2017
From Chai Wah Wu, Jul 29 2016: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
G.f.: 2*x*(1 - 2*x + 7*x^2)/(1 - x)^5. (End)

A130218 Partial sums of A100119. Sum of first n of the n-th centered n-gonal numbers.

Original entry on oeis.org

1, 3, 10, 29, 70, 146, 273, 470, 759, 1165, 1716, 2443, 3380, 4564, 6035, 7836, 10013, 12615, 15694, 19305, 23506, 28358, 33925, 40274, 47475, 55601, 64728, 74935, 86304, 98920, 112871, 128248, 145145, 163659, 183890, 205941, 229918, 255930

Offset: 1

Jonathan Vos Post, Aug 04 2007

This is to n-th centered n-gonal numbers (A100119) as A101357 is to n-th n-gonal numbers (A060354). a(2) = 3 and a(4) = 29 are primes. a(39) = 284089 = 13^2 * 41^2.

			a(41) = 1 + 2 + 7 + 19 + 41 + 76 + 127 + 197 + 289 + 406 + 551 + 727 + 937 + 1184 + 1471 + 1801 + 2177 + 2602 + 3079 + 3611 + 4201 + 4852 + 5567 + 6349 + 7201 + 8126 + 9127 + 10207 + 11369 + 12616 + 13951 + 15377 + 16897 + 18514 + 20231 + 22051 + 23977 + 26012 + 28159 + 30421 + 32801 + 35302 = 382613.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A060354, A100119, A101357.

LinearRecurrence[{5,-10,10,-5,1},{1,3,10,29,70},40] (* Harvey P. Dale, Jun 02 2018 *)

a(n) = (n*(26-3*n-2*n^2+3*n^3))/24. G.f.: x*(x^3-5*x^2+2*x-1) / (x-1)^5. - Colin Barker, Apr 29 2013

A296374 a(0) = 3; a(n) = a(n-1)(a(n-1)^2 - 3a(n-1) + 4)/2.

Original entry on oeis.org

3, 6, 66, 137346, 1295413937737986, 1086915296274625337063297033180803022465442306

Offset: 0

Ilya Gutkovskiy, Dec 11 2017

nonn

The next term is too large to include.

			a(0) = 3;
a(1) = 6 and 6 is the 3rd triangular number;
a(2) = 66 and 66 is the 6th hexagonal number;
a(3) = 137346 and 137346 is the 66th 66-gonal number, etc.

Index to sequences related to polygonal numbers

Cf. A001146, A007501, A057145, A060354, A097419, A099137, A099147, A099153.

RecurrenceTable[{a[0] == 3, a[n] == a[n - 1] (a[n - 1]^2 - 3 a[n - 1] + 4)/2}, a[n], {n, 5}]

a(0) = 3; a(n) = [x^a(n-1)] x*(1 - 2*x + 4*x^2)/(1 - x)^4.

a(0) = 3; a(n) = a(n-1)! * [x^a(n-1)] exp(x)*x*(1 + x^2/2).

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).

A330892 Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, -3, 1, 1, 0, -8, 0, 2, 1, 0, -15, -2, 3, 3, 1, 0, -24, -5, 4, 6, 4, 1, 0, -35, -9, 5, 10, 9, 5, 1, 0, -48, -14, 6, 15, 16, 12, 6, 1, 0, -63, -20, 7, 21, 25, 22, 15, 7, 1, 0, -80, -27, 8, 28, 36, 35, 28, 18, 8, 1, 0, -99, -35, 9, 36, 49, 51, 45, 34, 21, 9, 1, 0

Offset: 1

Robert G. Wilson v, Apr 27 2020

\c  0 1  2  3  4   5   6   7   8   9  10  11   12   13    14  15  ...
r\
_0  0 1  0 -3 -8 -15 -24 -35 -48 -63 -80 -99 -120 -143 -168 -195  A067998
_1  0 1  1  0 -2  -5  -9 -14 -20 -27 -35 -44  -54  -65  -77  -90  A080956
_2  0 1  2  3  4   5   6   7   8   9  10  11   12   13   14   15  A001477
_3  0 1  3  6 10  15  21  28  36  45  55  66   78   91  105  120  A000217
_4  0 1  4  9 16  25  36  49  64  81 100 121  144  169  196  225  A000290
_5  0 1  5 12 22  35  51  70  92 117 145 176  210  247  287  330  A000326
_6  0 1  6 15 28  45  66  91 120 153 190 231  276  325  378  435  A000384
_7  0 1  7 18 34  55  81 112 148 189 235 286  342  403  469  540  A000566
_8  0 1  8 21 40  65  96 133 176 225 280 341  408  481  560  645  A000567
_9  0 1  9 24 46  75 111 154 204 261 325 396  474  559  651  750  A001106
10  0 1 10 27 52  85 126 175 232 297 370 451  540  637  742  855  A001107
11  0 1 11 30 58  95 141 196 260 333 415 506  606  715  833  960  A051682
12  0 1 12 33 64 105 156 217 288 369 460 561  672  793  924 1065  A051624
13  0 1 13 36 70 115 171 238 316 405 505 616  738  871 1015 1170  A051865
14  0 1 14 39 76 125 186 259 344 441 550 671  804  949 1106 1275  A051866
15  0 1 15 42 82 135 201 280 372 477 595 726  870 1027 1197 1380  A051867
...
Each row has a second forward difference of (r-2) and each column has a forward difference of c(c-1)/2.

E. Deza and M. Deza, Figurate Numbers, World Scientific, 2012; see p. 45.
Eric Weisstein's World of Mathematics, Polygonal Number.
Wikipedia, Polygonal number.
Index to sequences related to polygonal numbers

Cf. A317302 (the same array) but read by ascending antidiagonals.
Sub-arrays: A089000, A139600, A206735;
Rows: A067998, A080956, A001477, A000217, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A051682, A051624, A051865, A051866, A051867, A051868, A051869, A051870, A051871, A051872, A051873, A051874, A051875, A051876, A255184, A255185, A255186, A161935, A255187, A254474, ..., ;
Columns (maybe missing some leading terms): A000004, A000012, A001477, A008585, A016957, A017329, A139606, A139607, A139608, A139609, A139610, A139611, A139612, A139613, A139614, A139615, A139616, A139617, A139618, A139619, A139620;
Diagonals: A256857, A127736, A002411, A006003, A006000, A064808, A060354, A162607, A077414,
Number of times k>1 appears: A129654, First occurrence of k: A063778.

Table[ PolygonalNumber[r - c, c], {r, 0, 11}, {c, r, 0, -1}] // Flatten

P(r, c) = (r - 2)(c(c-1)/2) + c.

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

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

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A366203 a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(3*n,n-k-1) * (n-3)^k.

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A117665 n times the n-th n-gonal number.

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A130218 Partial sums of A100119. Sum of first n of the n-th centered n-gonal numbers.

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A296374 a(0) = 3; a(n) = a(n-1)*(a(n-1)^2 - 3*a(n-1) + 4)/2.

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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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A330892 Square array of polygonal numbers read by descending antidiagonals (the transpose of A317302).

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A366203 a(n) = (1/n) * Sum_{k=0..n-1} binomial(n+k-1,k) * binomial(3n,n-k-1) (n-3)^k.

A296374 a(0) = 3; a(n) = a(n-1)(a(n-1)^2 - 3a(n-1) + 4)/2.

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.