A081436 -id:A081436 - cp's OEIS frontend

A157705 G.f.s of the z^p coefficients of the polynomials in the GF4 denominators of A156933.

1, 1, 3, 2, 18, 128, 171, 42, 1, 22, 1348, 11738, 26293, 17693, 3271, 115, 13, 6122, 228986, 2070813, 6324083, 7397855, 3361536, 544247, 24590, 155, 3, 17248, 2413434, 67035224, 612026240, 2274148882

Offset: 0

Johannes W. Meijer, Mar 07 2009

The formula for the PDGF4(z;n) polynomials in the GF4 denominators of A156933 can be found below.
The general structure of the GFKT4(z;p) that generate the z^p coefficients of the PDGF4(z;n) polynomials can also be found below. The KT4(z;p) polynomials in the numerators of the GFKT4(z;p) have a nice symmetrical structure.
The sequence of the number of terms of the first few KT4(z;p) polynomials is 1, 3, 5, 7, 10, 13, 15, 18, 20, 23, 26, 29, 32, 34, 37, 40, 42. The differences of this sequence and that of the number of terms of the KT3(z;p), see A157704, follow a simple pattern.
A Maple algorithm that generates relevant GFKT4(z;p) information can be found below.

			Some PDGF4 (z;n) are:
  PDGF4(z; n=3) = (1-7*z)*(1-5*z)^4*(1-3*z)^7*(1-z)^10
  PDGF4(z; n=4) = (1-9*z)*(1-7*z)^4*(1-5*z)^7*(1-3*z)^10*(1-z)^13
The first few GFKT4's are:
  GFKT4(z;p=0) = 1/(1-z)
  GFKT4(z;p=1) = -(1+3*z+2*z^2)/(1-z)^4
  GFKT4(z;p=2) = z*(18+128*z+171*z^2+42*z^3+z^4)/(1-z)^7
Some KT4(z,p) polynomials are:
  KT4(z;p=2) = 18+128*z+171*z^2+42*z^3+z^4
  KT4(z;p=3) = 22+1348*z+11738*z^2+26293*z^3+17693*z^4+3271*z^5+115*z^6

Originator sequence A156933.
See A081436 for the z^1 coefficients and A157708 for the z^2 coefficients.
Row sums equal A064350 and those of A157704.
Cf. A157702, A157703, A157704.

p:=2; fn:=sum((-1)^(n1+1)*binomial(3*p+1,n1) *a(n-n1),n1=1..3*p+1): fk:=rsolve(a(n) = fn,a(k)): for n2 from 0 to 3*p+1 do fz(n2):=product((1-(2*n2+1-(2*k))*z)^(3*k+1), k=0..n2): a(n2):= coeff(fz(n2),z,p): end do: b:=n-> a(n): seq(b(n), n=0..3*p+1); a(n)=fn; a(k)= sort (simplify(fk)); GFKT4(p):=sum((fk)*z^k,k=0..infinity); q4:=ldegree((numer (GFKT4(p)))): KT4(p):=sort((-1)^(p)*simplify((GFKT4(p)*(1-z)^(3*p+1))/z^q4),z, ascending);

PDGF4(z;n) = Product_{k=0..n} (1-(2*n+1-2*k)*z)^(3*k+1) with n = 1, 2, 3, ...
GFKT4(z;p) = (-1)^(p)*(z^q4)*KT4(z, p)/(1-z)^(3*p+1) with p = 0, 1, 2, ...
The recurrence relation for the z^p coefficients a(n) is a(n) = Sum_{k=1..3*p+1} (-1)^(k+1)*binomial(3*p + 1, k)*a(n-k) with p = 0, 1, 2, ... .

A241016 Triangle read by rows: T(n, k) = sum of k-th row of n X n square filled with the numbers 1 through n^2 reading across rows left-to-right.

Original entry on oeis.org

1, 3, 7, 6, 15, 24, 10, 26, 42, 58, 15, 40, 65, 90, 115, 21, 57, 93, 129, 165, 201, 28, 77, 126, 175, 224, 273, 322, 36, 100, 164, 228, 292, 356, 420, 484, 45, 126, 207, 288, 369, 450, 531, 612, 693, 55, 155, 255, 355, 455, 555, 655, 755, 855, 955, 66, 187, 308, 429, 550

Offset: 1

Kival Ngaokrajang, Aug 08 2014

See illustration in links.

The corresponding triangle with column sums is found in A251630. - Wolfdieter Lang, Dec 09 2014

			The triangle T(n, k) begins:
n\k  1   2   3   4   5   6   7   8   9  10 ...
1:   1
2:   3   7
3:   6  15  24
4:  10  26  42  58
5:  15  40  65  90 115
6:  21  57  93 129 165 201
7:  28  77 126 175 224 273 322
8:  36 100 164 228 292 356 420 484
9:  45 126 207 288 369 450 531 612 693
10: 55 155 255 355 455 555 655 755 855 955
... reformatted - _Wolfdieter Lang_, Dec 08 2014

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Kival Ngaokrajang, Illustration of initial terms

Columns k=1..6: A000217, A005449, A005475, A022265, A022267, A022269.
Diagonals: A081436, A059270, ...
Row sums: A037270.

Table[Sum[n*(k - 1) + j, {j,1,n}], {n,1,10}, {k,1,n}] // Flatten (* G. C. Greubel, Aug 23 2017 *)

trg(nn) = {for (n=1, nn, mm = matrix(n, n, i, j, j + n*(i-1)); for (i=1, n, print1(sum(j=1, n, mm[i, j]), ", ");); print(););} \\ Michel Marcus, Sep 15 2014

T(n, k) = Sum_{j=1..n} (n*(k-1)+ j), for n >= k >= 1. See the Michel Marcus program. - Wolfdieter Lang, Dec 08 2014

T(n, k) = binomial(n+1, 2) + n^2*(k-1). - Wolfdieter Lang, Dec 09 2014

Edited. - Wolfdieter Lang, Dec 08 2014

A005920 Tricapped prism numbers.

Original entry on oeis.org

1, 9, 33, 82, 165, 291, 469, 708, 1017, 1405, 1881, 2454, 3133, 3927, 4845, 5896, 7089, 8433, 9937, 11610, 13461, 15499, 17733, 20172, 22825, 25701, 28809, 32158, 35757, 39615, 43741, 48144, 52833, 57817, 63105, 68706, 74629, 80883, 87477, 94420

Offset: 0

N. J. A. Sloane

a(n) = (n+1)*A000326(n+1) - Sum_{i=0...n} A001477(i) = (n+1)*((n+1)*(3*n+2)/2) - A000217(n) = (n+1)*(3*n^2+4n+2)/2. - Bruno Berselli, Apr 25 2010

Also central terms of triangle A093445: a(n) = A093445(2*n+1,n+1). - Reinhard Zumkeller, Oct 03 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. for recursive method [Ar(m) is the m-th term of a sequence in the OEIS] a(n) = n*Ar(n) - A000217(n-1) or a(n) = (n+1)*Ar(n+1) - A000217(n) or similar: A081436, A005945, A006003 and the terms T(2, n) or T(3, n) in the sequence A125860. - Bruno Berselli, Apr 25 2010

Cf. A000326, A001477, A093445.

a005920 n = (n * (n * (3 * n + 7) + 6) + 2) `div` 2
-- Reinhard Zumkeller, Oct 03 2012

[(3*n^3+7*n^2+6*n+2)/2 : n in [0..50]]; // Wesley Ivan Hurt, May 05 2021

a:=n->(3*n^3+7*n^2+6*n+2)/2: seq(a(n),n=0..60);
A005920:=(1+5*z+3*z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation

CoefficientList[ Series[(1+5x+3x^2)/(1-x)^4, {x, 0, 39}], x] (* Jean-François Alcover, Dec 02 2011, after Simon Plouffe *)
LinearRecurrence[{4,-6,4,-1},{1,9,33,82},40] (* Harvey P. Dale, Sep 25 2012 *)

a(n)=n*(3*n^2+7*n+6)/2+1 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (1/2) * (3*n^3 + 7*n^2 + 6*n + 2). - Ralf Stephan, Apr 20 2004
a(0)=1, a(1)=9, a(2)=33, a(3)=82, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Sep 25 2012
E.g.f.: exp(x)*(2 + 16*x + 16*x^2 + 3*x^3)/2. - Stefano Spezia, Jun 10 2022

More terms from Emeric Deutsch, May 09 2004

A132117 Binomial transform of [1, 7, 17, 17, 6, 0, 0, 0, ...].

Original entry on oeis.org

1, 8, 32, 90, 205, 406, 728, 1212, 1905, 2860, 4136, 5798, 7917, 10570, 13840, 17816, 22593, 28272, 34960, 42770, 51821, 62238, 74152, 87700, 103025, 120276, 139608, 161182, 185165, 211730, 241056, 273328, 308737, 347480, 389760, 435786, 485773, 539942, 598520

Offset: 1

Gary W. Adamson, Aug 10 2007

Equals row sums of triangle A178067. - Gary W. Adamson, May 18 2010
Antidiagonal sums of the convolution array A213771. - Clark Kimberling, Jul 04 2012
Partial sums of A081436. - J. M. Bergot, Jun 20 2013

			a(3) = 32 = (1, 2, 1) dot (1, 7, 17) = (1 + 14 + 17).
a(5) = 15^2 - (10+6+3+1) = A000537(5) - A000292(4) = 225 - 20 = 205. - _Bruno Berselli_, May 01 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A178067. - Gary W. Adamson, May 18 2010

a:= n-> (Matrix([[0,0,2,13,46]]). Matrix(5, (i,j)-> if (i=j-1) then 1 elif j=1 then [5,-10,10,-5,1][i] else 0 fi)^n)[1,1]: seq(a(n), n=1..29); # Alois P. Heinz, Aug 07 2008
a:= n-> (4+(6+(8+6*n)*n)*n)*n/24: seq(a(n),n=1..40); # Alois P. Heinz, Aug 07 2008

Table[(4 n + 6 n^2 + 8 n^3 + 6 n^4) / 24, {n, 50}] (* Vincenzo Librandi, Jun 21 2013 *)

a(n) = (4*n+6*n^2+8*n^3+6*n^4)/24 \\ Charles R Greathouse IV, Sep 03 2011

Let M = the infinite lower triangular matrix of the natural numbers: [1; 2,3; 4,5,6; ...]; and V = [1, 2, 3, ...]. Then M*V = A132117.
O.g.f.: -x(1+x)(2x+1)/(-1+x)^5. - R. J. Mathar, Apr 02 2008
a(n) = (4*n + 6*n^2 + 8*n^3 + 6*n^4)/24. - Alois P. Heinz, Aug 07 2008
a(n) = A000217(n)^2 - Sum_{i=1..n-1} A000217(i) = n*(n+1)*(3*n^2+n+2)/12. - Bruno Berselli, May 01 2010

More terms from R. J. Mathar, Apr 02 2008

A213831 Rectangular array: (row n) = b**c, where b(h) = 2h-1, c(h) = 3n-5+3*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 7, 4, 24, 19, 7, 58, 51, 31, 10, 115, 106, 78, 43, 13, 201, 190, 154, 105, 55, 16, 322, 309, 265, 202, 132, 67, 19, 484, 469, 417, 340, 250, 159, 79, 22, 693, 676, 616, 525, 415, 298, 186, 91, 25, 955, 936, 868, 763, 633

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A213832.
Antidiagonal sums: A212560.
row 1, (1,3,5,7,...)**(1,4,7,10,...): A081436.
Row 2, (1,3,5,7,...)**(4,7,10,13,...): A162254.
Row 3, (1,3,5,7,...)**(7,10,13,16,...): (2*k^3 + 11*k^2 + k)/2.
For a guide to related arrays, see A212500.

			1....7....24....58....115
4....19...51....106...190
7....31...78....154...265
10...43...105...202...340
13...55...132...250...415

Cf. A212500

b[n_]:=2n-1;c[n_]:=3n-2;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213831 *)
Table[t[n,n],{n,1,40}] (* A213832 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A212560 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).
G.f. for row n: f(x)/g(x), where f(x) = x*((3*n-2) + 3*x - (3*n-5)*x^2) and g(x) = (1-x)^4.
Northwest corner (the array is read by falling antidiagonals):

A213833 Rectangular array: (row n) = b**c, where b(h) = 3h-2, c(h) = 2n-3+2*h, n>=1, h>=1, and ** = convolution.

Original entry on oeis.org

1, 7, 3, 24, 17, 5, 58, 48, 27, 7, 115, 102, 72, 37, 9, 201, 185, 146, 96, 47, 11, 322, 303, 255, 190, 120, 57, 13, 484, 462, 405, 325, 234, 144, 67, 15, 693, 668, 602, 507, 395, 278, 168, 77, 17, 955, 927, 852, 742, 609, 465

Offset: 1

Clark Kimberling, Jul 04 2012

Principal diagonal: A103748.
Antidiagonal sums: A213834.
Row 1, (1,3,5,7,...)**(1,3,5,7,...): A081436.
Row 2, (1,3,5,7,...)**(3,5,7,9,...): A144640.
Row 3, (1,3,5,7,...)**(5,7,9,11,...): (2*k^3 + 11*k^2 - 3*k)/2.
For a guide to related arrays, see A212500.

			Northwest corner (the array is read by falling antidiagonals):
1....7....24....58....115
3....17...48....102...185
5....27...72....146...255
7....37...96....190...325
9....47...120...234...395
11...57...144...278...465

Clark Kimberling, Antidiagonals n = 1..12, flattened

Cf. A212500.

b[n_]:=3n-2;c[n_]:=2n-1;
t[n_,k_]:=Sum[b[k-i]c[n+i],{i,0,k-1}]
TableForm[Table[t[n,k],{n,1,10},{k,1,10}]]
Flatten[Table[t[n-k+1,k],{n,12},{k,n,1,-1}]]
r[n_]:=Table[t[n,k],{k,1,60}] (* A213833 *)
Table[t[n,n],{n,1,40}] (* A130748 *)
s[n_]:=Sum[t[i,n+1-i],{i,1,n}]
Table[s[n],{n,1,50}] (* A213834 *)

T(n,k) = 4*T(n,k-1)-6*T(n,k-2)+4*T(n,k-3)-T(n,k-4).

G.f. for row n: f(x)/g(x), where f(x) = x*((2*n-1) + (2*n+1)*x - (4*n-6)*x^2) and g(x) = (1-x)^4.

A081438 Diagonal in array of n-gonal numbers A081422.

Original entry on oeis.org

1, 11, 36, 82, 155, 261, 406, 596, 837, 1135, 1496, 1926, 2431, 3017, 3690, 4456, 5321, 6291, 7372, 8570, 9891, 11341, 12926, 14652, 16525, 18551, 20736, 23086, 25607, 28305, 31186, 34256, 37521, 40987, 44660, 48546, 52651, 56981, 61542, 66340

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081436, A081437, A081441, A177058.

List([0..45], n-> (2*n^3+9*n^2+9*n+2)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3+9*n^2+9*n+2)/2: n in [0..45]]; // Vincenzo Librandi, Aug 08 2013

seq((2*n^3+9*n^2+9*n+2)/2, n=0..45); # G. C. Greubel, Aug 14 2019

CoefficientList[Series[(1 +6x -9x^2 +2x^3)/(1-x)^5, {x, 0, 45}], x] (* Vincenzo Librandi, Aug 08 2013 *)
LinearRecurrence[{4,-6,4,-1},{1,11,36,82},50] (* Harvey P. Dale, Jan 20 2022 *)

vector(45, n, n--; (2*n^3+9*n^2+9*n+2)/2) \\ G. C. Greubel, Aug 14 2019

[(2*n^3+9*n^2+9*n+2)/2 for n in (0..45)] # G. C. Greubel, Aug 14 2019

a(n) = (2*n^3+9*n^2+9*n+2)/2.
G.f.: (1+6*x-9*x^2+2*x^3)/(1-x)^5.
From Bruno Berselli, Jun 04 2010: (Start)
G.f.: (1+7*x-2*x^2)/(1-x)^4 (simplified).
a(n) = (n+1)*(2*n^2+7*n+2)/2.
a(n) -4*a(n-1) +6*a(n-2) -4*a(n-3) +a(n-4) = 0, with n>3.
a(n) = (A177058(n+3) + A177058(n+2))/2. (End)
E.g.f.: (1/2)*exp(x)*(2 +20*x + 15*x^2 + 2*x^3). - Stefano Spezia, Aug 15 2019

A300401 Array T(n,k) = n(binomial(k, 2) + 1) + k(binomial(n, 2) + 1) read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 7, 8, 7, 4, 5, 11, 14, 14, 11, 5, 6, 16, 22, 24, 22, 16, 6, 7, 22, 32, 37, 37, 32, 22, 7, 8, 29, 44, 53, 56, 53, 44, 29, 8, 9, 37, 58, 72, 79, 79, 72, 58, 37, 9, 10, 46, 74, 94, 106, 110, 106, 94, 74, 46, 10, 11, 56, 92, 119

Offset: 0

Franck Maminirina Ramaharo, Mar 05 2018

Antidiagonal sums are given by 2*A055795.
Rows/columns n are binomial transform of {n, A152947(n+1), n, 0, 0, 0, ...}.
Some primes in the array are
n = 1: {2, 7, 11, 29, 37, 67, 79, 137, 191, 211, 277, 379, ...} = A055469, primes of the form k*(k + 1)/2 + 1;
n = 3: {3, 7, 37, 53, 479, 653, 1249, 1619, 2503, 3727, 4349, 5737, 7109, 8179, 9803, 11839, 12107, ...};
n = 4: {11, 37, 79, 137, 211, 821, 991, 1597, 1831, 2081, 2347, ...} = A188382, primes of the form 8*(2*k - 1)^2 + 2*(2*k - 1) + 1.

			The array T(n,k) begins
0     1    2    3    4     5     6     7     8     9    10    11  ...
1     2    4    7   11    16    22    29    37    46    56    67  ...
2     4    8   14   22    32    44    58    74    92   112   134  ...
3     7   14   24   37    53    72    94   119   147   178   212  ...
4    11   22   37   56    79   106   137   172   211   254   301  ...
5    16   32   53   79   110   146   187   233   284   340   401  ...
6    22   44   72  106   146   192   244   302   366   436   512  ...
7    29   58   94  137   187   244   308   379   457   542   634  ...
8    37   74  119  172   233   302   379   464   557   658   767  ...
9    46   92  147  211   284   366   457   557   666   784   911  ...
10   56  112  178  254   340   436   542   658   784   920  1066  ...
11   67  134  212  301   401   512   634   767   911  1066  1232  ...
12   79  158  249  352   467   594   733   884  1047  1222  1409  ...
13   92  184  289  407   538   682   839  1009  1192  1388  1597  ...
14  106  212  332  466   614   776   952  1142  1346  1564  1796  ...
15  121  242  378  529   695   876  1072  1283  1509  1750  2006  ...
16  137  274  427  596   781   982  1199  1432  1681  1946  2227  ...
17  154  308  479  667   872  1094  1333  1589  1862  2152  2459  ...
18  172  344  534  742   968  1212  1474  1754  2052  2368  2702  ...
19  191  382  592  821  1069  1336  1622  1927  2251  2594  2956  ...
20  211  422  653  904  1175  1466  1777  2108  2459  2830  3221  ...
...
The inverse binomial transforms of the columns are
0     1    2    3    4     5     6     7     8     9    10    11  ...  A001477
1     1    2    4    7    11    22    29    37    45    56    67  ...  A152947
0     1    2    3    4     5     6     7     8     9    10    11  ...  A001477
0     0    0    0    0     0     0     0     0     0     0     0  ...
0     0    0    0    0     0     0     0     0     0     0     0  ...
0     0    0    0    0     0     0     0     0     0     0     0  ...
...

Miklós Bóna, Introduction to Enumerative Combinatorics, McGraw-Hill, 2007.
L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel Publishing Company, 1974.
R. P. Stanley, Enumerative Combinatorics, second edition, Cambridge University Press, 2011.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.

Cf. A000124, A001477, A006000, A008815, A014206, A051601, A055469, A077028, A081436, A084849, A131074, A134394, A139600, A141387, A179000, A188377, A188382, A273465.

T := (n, k) -> n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1);
for n from 0 to 20 do seq(T(n, k), k = 0 .. 20) od;

T[n_, k_] := n (Binomial[k, 2] + 1) + k (Binomial[n, 2] + 1);
Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 07 2018 *)

T(n, k) := n*(binomial(k, 2) + 1) + k*(binomial(n, 2) + 1)$
for n:0 thru 20 do
  print(makelist(T(n, k), k, 0, 20));

T(n, k) = n*(binomial(k,2) + 1) + k*(binomial(n,2) + 1);
tabl(nn) = for (n=0, nn, for (k=0, nn, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 12 2018

T(n,k) = T(k,n) = n*A152947(k+1) + k*A152947(n+1).
T(n,0) = A001477(n).
T(n,1) = A000124(n).
T(n,2) = A014206(n).
T(n,3) = A273465(3*n+2).
T(n,4) = A084849(n+1).
T(n,n) = A179000(n-1,n), n >= 1.
T(2*n,2*n) = 8*A081436(n-1), n >= 1.
T(2*n+1,2*n+1) = 2*A006000(2*n+1).
T(n,n+1) = A188377(n+3).
T(n,n+2) = A188377(n+2), n >= 1.
Sum_{k=0..n} T(k,n-k) = 2*(binomial(n, 4) + binomial(n, 2)).
G.f.: -((2*x*y - y - x)*(2*x*y - y - x + 1))/(((x - 1)*(y - 1))^3).
E.g.f.: (1/2)*(x + y)*(x*y + 2)*exp(x + y).

A081423 Subdiagonal of array of n-gonal numbers A081422.

Original entry on oeis.org

1, 3, 12, 34, 75, 141, 238, 372, 549, 775, 1056, 1398, 1807, 2289, 2850, 3496, 4233, 5067, 6004, 7050, 8211, 9493, 10902, 12444, 14125, 15951, 17928, 20062, 22359, 24825, 27466, 30288, 33297, 36499, 39900, 43506, 47323, 51357, 55614, 60100

Offset: 0

Paul Barry, Mar 21 2003

One of a family of sequences with palindromic generators.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A081435, A081436, A081437.

List([0..40], n-> (2*n^3+n^2+n+2)/2); # G. C. Greubel, Aug 14 2019

[(2*n^3+n^2+n+2)/2: n in [0..40]]; // Vincenzo Librandi, Aug 08 2013

a := n-> (2*n^3+n^2+n+2)/2; seq(a(n), n = 0..40); # G. C. Greubel, Aug 14 2019

CoefficientList[Series[(1 -2x +7x^2 -6x^3)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Aug 08 2013 *)

vector(40, n, n--; (2*n^3+n^2+n+2)/2) \\ G. C. Greubel, Aug 14 2019

[(2*n^3+n^2+n+2)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019

a(n) = (2*n^3 + n^2 + n + 2)/2.
G.f.: (1 -2*x +7*x^2 -6*x^3)/(1-x)^5.
E.g.f.: (2 +4*x +7*x^2 +2*x^3)*exp(x)/2. - G. C. Greubel, Aug 14 2019

A244418 Triangle read by rows T(n,m) = nm +(n-1)(m-1), for n >= m >= 1.

Original entry on oeis.org

1, 2, 5, 3, 8, 13, 4, 11, 18, 25, 5, 14, 23, 32, 41, 6, 17, 28, 39, 50, 61, 7, 20, 33, 46, 59, 72, 85, 8, 23, 38, 53, 68, 83, 98, 113, 9, 26, 43, 60, 77, 94, 111, 128, 145, 10, 29, 48, 67, 86, 105, 124, 143, 162, 181, 11, 32, 53, 74, 95, 116, 137, 158, 179, 200, 221

Offset: 1

Wolfdieter Lang, Jul 10 2014

This table is motivated by an entry of Aki Halme (A243907); see also A053726. a(n,m) is the number of stars on an array similar to the one appearing on the flag of the United States with n columns of m stars interchanged with (n-1) columns of (m-1) stars, for n>=2 and m = 2, ..., n.
The column sequences of the rectangular array R(n,m) = n*m + (n-1)*(m-1) = (2*n-1)*(m-1) + n for n >= 1 and m >= 1 (just symmetrize the given triangular array) are congruent n (mod (2*n-1)), n >= 1. With the odd modulus M = 2*n-1 and for M = d*L, that is d|n and L = (2*n-1)/d one can derive an identity for R(n,m) = d*(L*(m-1) + x) + (n - x*d) = d*k + (d+1)/2 (new modulus d) with k = L*(m-1) + x and n - x*d = (d+1)/2, that is x = ((2*n-1) - d)/(2*d) = (L-1)/2 which is a positive integer because L is odd. Then k = (2*L*m - (L+1))/2, also an integer. Thus for each divisor d of n the identity R(n, m) = R((d+1)/2, k+1) = R((d+1)/2, ((2*m-1)*(2*n-1)/d + 1)/2) holds.
The preceding identity shows that for odd composite moduli M = 2*n - 1 (with nontrivial divisors d of M) the sequence R(n,m), m >= 1 is a subsequence of the one for each modulus d. For example, for M = 15 = 3*5, n = 8, 15*(m-1) + 8 = 3*(5*m-3) + 2 = 5*(3*m-2) + 3 for m >= 1.

			The triangle T(n,m) begins:
n\m   1  2  3   4   5   6   7   8   9  10 ...
1:    1
2:    2  5
3:    3  8 13
4:    4 11 18  25
5:    5 14 23  32  41
6:    6 17 28  39  50  61
7:    7 20 33  46  59  72  85
8:    8 23 38  53  68  83  98 113
9:    9 26 43  60  77  94 111 128 145
10:  10 29 48  67  86 105 124 143 162 181 ...
For more rows see the link.

Wolfdieter Lang, First twenty rows of the triangle.

Cf. A243907, A144650, A053726, A081436 (row sums).

tabl(nn) = {for (n=1, nn, for (m=1, n, print1(n*m + (n-1)*(m-1), ", ");); print(););} \\ Michel Marcus, Jan 11 2015

T(n,m) = n*m + (n-1)*(m-1) = (2*n-1)*(m-1) + n, for n>=m, else 0.
G.f. for column m: G(m, x) = x^m*((2*m^2 - 2*m + 1) - 2*(m - 1)^2*x)/(1 - x)^2.
G.f. for triangle: sum(n >= 1, sum(m = 1..n, T(n,m)*x^m*y^n ) ) = (x*y+1)*(2*x*y^2-x*y-1)*x*y/((-1+y)^2*(x*y-1)^3). - Robert Israel, Jan 11 2015

cp's OEIS Frontend

A157705 G.f.s of the z^p coefficients of the polynomials in the GF4 denominators of A156933.

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A241016 Triangle read by rows: T(n, k) = sum of k-th row of n X n square filled with the numbers 1 through n^2 reading across rows left-to-right.

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A005920 Tricapped prism numbers.

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A132117 Binomial transform of [1, 7, 17, 17, 6, 0, 0, 0, ...].

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A213831 Rectangular array: (row n) = b**c, where b(h) = 2*h-1, c(h) = 3*n-5+3*h, n>=1, h>=1, and ** = convolution.

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A213833 Rectangular array: (row n) = b**c, where b(h) = 3*h-2, c(h) = 2*n-3+2*h, n>=1, h>=1, and ** = convolution.

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A081438 Diagonal in array of n-gonal numbers A081422.

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A213831 Rectangular array: (row n) = b**c, where b(h) = 2h-1, c(h) = 3n-5+3*h, n>=1, h>=1, and ** = convolution.

A213833 Rectangular array: (row n) = b**c, where b(h) = 3h-2, c(h) = 2n-3+2*h, n>=1, h>=1, and ** = convolution.

A300401 Array T(n,k) = n(binomial(k, 2) + 1) + k(binomial(n, 2) + 1) read by antidiagonals.

A244418 Triangle read by rows T(n,m) = nm +(n-1)(m-1), for n >= m >= 1.