A063521 -id:A063521 - cp's OEIS frontend

A004126 a(n) = n(7n^2 - 1)/6.

0, 1, 9, 31, 74, 145, 251, 399, 596, 849, 1165, 1551, 2014, 2561, 3199, 3935, 4776, 5729, 6801, 7999, 9330, 10801, 12419, 14191, 16124, 18225, 20501, 22959, 25606, 28449, 31495, 34751, 38224, 41921, 45849, 50015, 54426, 59089, 64011

Offset: 0

Albert D. Rich (Albert_Rich(AT)msn.com)

3-dimensional analog of centered polygonal numbers.
Sum of n triangular numbers starting from T(n), where T = A000217. E.g., a(4) = T(4) + T(5) + T(6) + T(7) = 10 + 15 + 21 + 28 = 74. - Amarnath Murthy, Jul 16 2004
Also as a(n) = (1/6)*(7*n^3-n), n>0: structured heptagonal diamond numbers (vertex structure 8). Cf. A100179 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Partial sums of A069099, centered heptagonal numbers (A000566). - Jonathan Vos Post, Mar 16 2006
Binomial transform of (0, 1, 7, 7, 0, 0, 0, ...) and third partial sum of (0, 1, 6, 7, 7, 7, ...). - Gary W. Adamson, Oct 05 2015

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.
Cf. A000217, A000566, A016993, A069099.
Cf. A000447, A000292.

[n*(7*n^2-1)/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011

seq(binomial(2*n+1,3)-binomial(n+1,3), n=0..38); # Zerinvary Lajos, Jan 21 2007

Table[n (7 n^2 - 1)/6, {n, 0, 80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

makelist(n*(7*n^2-1)/6,n,0,30); /* Martin Ettl, Jan 08 2013 */

vector(100, n, n--; n*(7*n^2 - 1)/6) \\ Altug Alkan, Oct 06 2015

a(n) = C(2*n+1,3)-C(n+1,3), n>=0. - Zerinvary Lajos, Jan 21 2007
a(n) = A000447(n) - A000292(n). - Zerinvary Lajos, Jan 21 2007
G.f.: x*(1+5*x+x^2)/(1-x)^4. - Colin Barker, Mar 02 2012
E.g.f.: (x/6)*(7*x^2 + 21*x + 6)*exp(x). - G. C. Greubel, Oct 05 2015
a(n) = Sum_{i = n..2*n-1} A000217(i). - Bruno Berselli, Sep 06 2017
a(n) = n^3 + Sum_{k=0..n-1} k*(k+1)/2. Alternately, a(n) = A000578(n) + A000292(n-1) for n>0. - Bruno Berselli, May 23 2018

A063522 a(n) = n(5n^2 - 3)/2.

Original entry on oeis.org

0, 1, 17, 63, 154, 305, 531, 847, 1268, 1809, 2485, 3311, 4302, 5473, 6839, 8415, 10216, 12257, 14553, 17119, 19970, 23121, 26587, 30383, 34524, 39025, 43901, 49167, 54838, 60929, 67455, 74431, 81872, 89793, 98209, 107135, 116586, 126577, 137123, 148239, 159940

Offset: 0

N. J. A. Sloane, Aug 02 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

(1/12)*t*(n^3 - n) + n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Bisections: A160674, A160699.

[n*(5*n^2 -3)/2: n in [0..30]]; // G. C. Greubel, May 02 2018

lst={};Do[AppendTo[lst, LegendreP[3, n]], {n, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 11 2008 *)
CoefficientList[Series[x*(1 + 13*x + x^2)/(1-x)^4, {x, 0, 50}], x] (* G. C. Greubel, Sep 01 2017 *)
LinearRecurrence[{4,-6,4,-1},{0,1,17,63},40] (* Harvey P. Dale, Sep 06 2023 *)

a(n) = { n*(5*n^2 - 3)/2 } \\ Harry J. Smith, Aug 25 2009

G.f.: x*(1 + 13*x + x^2)/(1-x)^4. - Colin Barker, Jan 10 2012

E.g.f.: (x/2)*(2 + 15*x + 5*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A004467 a(n) = n(11n^2 - 5)/6.

Original entry on oeis.org

0, 1, 13, 47, 114, 225, 391, 623, 932, 1329, 1825, 2431, 3158, 4017, 5019, 6175, 7496, 8993, 10677, 12559, 14650, 16961, 19503, 22287, 25324, 28625, 32201, 36063, 40222, 44689, 49475, 54591, 60048

Offset: 0

Albert D. Rich (Albert_Rich(AT)msn.com)

3-dimensional analog of centered polygonal numbers, that is: centered hendecagonal pyramidal numbers (see Deza paper in References).

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

[n*(11*n^2-5)/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011

Table[n(11n^2-5)/6,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,1,13,47},80] (* Harvey P. Dale, Sep 22 2013 *)

a(n)=n*(11*n^2-5)/6 \\ Charles R Greathouse IV, Sep 28 2011

G.f.: x*(1+9*x+x^2)/(1-x)^4. - Colin Barker, Jan 08 2012
a(0)=0, a(1)=1, a(2)=13, a(3)=47; for n>3, a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Harvey P. Dale, Sep 22 2013
E.g.f.: (x/6)*(6 + 33*x + 11*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A062025 a(n) = n(13n^2 - 7)/6.

Original entry on oeis.org

0, 1, 15, 55, 134, 265, 461, 735, 1100, 1569, 2155, 2871, 3730, 4745, 5929, 7295, 8856, 10625, 12615, 14839, 17310, 20041, 23045, 26335, 29924, 33825, 38051, 42615, 47530, 52809, 58465, 64511, 70960, 77825, 85119, 92855, 101046, 109705, 118845, 128479, 138620, 149281

Offset: 0

N. J. A. Sloane, Aug 02 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Table[n(13n^2-7)/6,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

a(n) = n*(13*n^2 - 7)/6 \\ Harry J. Smith, Jul 29 2009

From G. C. Greubel, Sep 01 2017: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: (x + 11*x^2 + x^3)/(1 - x)^4.
E.g.f.: (x/6)*(6 + 39*x + 13*x^2)*exp(x). (End)

A063523 a(n) = n(8n^2 - 5)/3.

Original entry on oeis.org

0, 1, 18, 67, 164, 325, 566, 903, 1352, 1929, 2650, 3531, 4588, 5837, 7294, 8975, 10896, 13073, 15522, 18259, 21300, 24661, 28358, 32407, 36824, 41625, 46826, 52443, 58492, 64989, 71950, 79391, 87328, 95777, 104754, 114275, 124356, 135013, 146262, 158119, 170600

Offset: 0

N. J. A. Sloane, Aug 02 2001

Also as a(n)=(1/6)*(16*n^3-10*n), n>0: structured octagonal anti-diamond numbers (vertex structure 17) (Cf. A100187 = alternate vertex; A100188 = structured anti-diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004

Harry J. Smith, Table of n, a(n) for n = 0..1000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

1/12*t*(n^3-n)+n for t = 2, 4, 6, ... gives A004006, A006527, A006003, A005900, A004068, A000578, A004126, A000447, A004188, A004466, A004467, A007588, A062025, A063521, A063522, A063523.

Table[n(8n^2-5)/3,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,1,18,67},81] (* or *) CoefficientList[ Series[ (x+14 x^2+x^3)/(x-1)^4,{x,0,80}],x] (* Harvey P. Dale, Jul 11 2011 *)

a(n) = n*(8*n^2 - 5)/3 \\ Harry J. Smith, Aug 25 2009

a(0)=0, a(1)=1, a(2)=18, a(3)=67, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)- a(n-4). - Harvey P. Dale, Jul 11 2011
G.f.: (x+14*x^2+x^3)/(x-1)^4. - Harvey P. Dale, Jul 11 2011
E.g.f.: (x/3)*(3 + 24*x + 8*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A100188 Polar structured meta-anti-diamond numbers, the n-th number from a polar structured n-gonal anti-diamond number sequence.

Original entry on oeis.org

1, 6, 27, 84, 205, 426, 791, 1352, 2169, 3310, 4851, 6876, 9477, 12754, 16815, 21776, 27761, 34902, 43339, 53220, 64701, 77946, 93127, 110424, 130025, 152126, 176931, 204652, 235509, 269730, 307551, 349216

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

			There are no 1- or 2-gonal anti-diamonds, so 1 and (2n+2) are the first and second terms since all the sequences begin as such.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A000578, A000447, A004466, A007588, A063521, A062523 - "polar" structured anti-diamonds; A100189 - "equatorial" structured meta-anti-diamond numbers; A006484 for other structured meta numbers; and A100145 for more on structured numbers.

[(1/6)*(2*n^4-2*n^2+6*n): n in [1..40]]; // Vincenzo Librandi, Aug 18 2011

Table[(2n^4-2n^2+6n)/6,{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1}, {1,6,27,84,205},40] (* Harvey P. Dale, May 11 2016 *)

vector(40, n, (n^4 -n^2 +3*n)/3) \\ G. C. Greubel, Nov 08 2018

a(n) = (1/6)*(2*n^4 - 2*n^2 + 6*n).
G.f.: x*(1 + x + 7*x^2 - x^3)/(1-x)^5. - Colin Barker, Apr 16 2012
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(1)=1, a(2)=6, a(3)=27, a(4)=84, a(5)=205. - Harvey P. Dale, May 11 2016
E.g.f.: (3*x + 6*x^2 + 6*x^3 + x^4)*exp(x)/3. - G. C. Greubel, Nov 08 2018

A166343 Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)p(n, x)/x, p(n, x) = xD( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x(1+x)/(1-x)^3, and p(3, x) = x(1+12*x+x^2)/(1-x)^4, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 27, 27, 1, 1, 58, 162, 58, 1, 1, 121, 718, 718, 121, 1, 1, 248, 2759, 5744, 2759, 248, 1, 1, 503, 9765, 36771, 36771, 9765, 503, 1, 1, 1014, 32816, 205674, 367710, 205674, 32816, 1014, 1, 1, 2037, 106560, 1052408, 3072594, 3072594, 1052408, 106560, 2037, 1

Offset: 1

Roger L. Bagula, Oct 12 2009

			Triangle begins as:
  1;
  1,    1;
  1,   12,      1;
  1,   27,     27,       1;
  1,   58,    162,      58,       1;
  1,  121,    718,     718,     121,       1;
  1,  248,   2759,    5744,    2759,     248,       1;
  1,  503,   9765,   36771,   36771,    9765,     503,      1;
  1, 1014,  32816,  205674,  367710,  205674,   32816,   1014,    1;
  1, 2037, 106560, 1052408, 3072594, 3072594, 1052408, 106560, 2037, 1;

Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Cf. A166340, A166341, A166344, A166345, A166346, A166349.

Cf. A063521, A123125.

(* First program *)
p[x_, 1]:= x/(1-x)^2;
p[x_, 2]:= x*(1+x)/(1-x)^3;
p[x_, 3]:= x*(1+12*x+x^2)/(1-x)^4;
p[x_, n_]:= p[x, n]= x*D[p[x, n-1], x]
Table[CoefficientList[(1-x)^(n+1)*p[x, n]/x, x], {n,12}]//Flatten
(* Second program *)
b[n_, k_, m_]:= If[n<2, 1, If[k==0, 0, k^(n-1)*((m+3)*k^2 - m)/3]];
t[n_, k_, m_]:= t[n,k,m]= Sum[(-1)^(k-j)*Binomial[n+1, k-j]*b[n,j,m], {j,0,k}];
T[n_, k_, m_]:= T[n,k,m]= If[k==1, 1, t[n-1,k,m] - t[n-1,k-1,m]];
Table[T[n,k,4], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 11 2022 *)

def b(n,k,m):
    if (n<2): return 1
    elif (k==0): return 0
    else: return k^(n-1)*((m+3)*k^2 - m)/3
@CachedFunction
def t(n,k,m): return sum( (-1)^(k-j)*binomial(n+1, k-j)*b(n,j,m) for j in (0..k) )
def A166343(n,k): return 1 if (k==1) else t(n-1,k,4) - t(n-1,k-1,4)
flatten([[A166343(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Mar 11 2022

T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+12*x+x^2)/(1-x)^4.
From G. C. Greubel, Mar 11 2022: (Start)
T(n, k) = t(n-1, k) - t(n-1, k-1), T(n,1) = 1, where t(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n+1, k-j)*b(n, j), b(n, k) = k^(n-2)*A063521(k), b(n, 0) = 1, and b(1, k) = 1.
T(n, n-k) = T(n, k). (End)

Edited by G. C. Greubel, Mar 11 2022

A245826 Triangle read by rows: T(m,n) is the Szeged index of the grid graph P_m X P_n (1 <= n <= m).

Original entry on oeis.org

0, 1, 16, 4, 59, 216, 10, 144, 526, 1280, 20, 285, 1040, 2530, 5000, 35, 496, 1809, 4400, 8695, 15120, 56, 791, 2884, 7014, 13860, 24101, 38416, 84, 1184, 4316, 10496, 20740, 36064, 57484, 86016, 120, 1689, 6156, 14970, 29580, 51435, 81984, 122676, 174960, 165, 2320, 8455, 20560, 40625, 70640, 112595, 168480, 240285, 330000

Offset: 1

Emeric Deutsch, Aug 06 2014

T(n,1) = Szeged index of the path tree P_n = A000292(n-1).
T(n,2) = Szeged index of the ladder graph P_2 X P_n = A063521(n).
T(n,3) = Szeged index of the grid graph P_3 X P_n = A245827(n).
T(n,n) = Szeged index of the grid graph P_n X P_n = A245828(n).

			T(2,2) = 16 because P_2 X P_2 is the square C_4 and each of its 4 edges contributes 2*2=4 to its Szeged index.
Triangle starts:
0;
1,16;
4,59,216;
10,144,526,1280;
20,285,1040,2530,5000;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
S. Klavzar, A. Rajapakse, I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett., 9, 1996, 45-49.

Cf. A000292, A063521, A245827, A245828.

Cf. A245940 (row sums), A245941 (central terms).

a245826 n k = n * k * (2 * n^2 * k^2 - n^2 - k^2) `div` 6
a245826_row n = map (a245826 n) [1..n]
a245826_tabl = map a245826_row [1..]
-- Reinhard Zumkeller, Aug 07 2014

T:=proc(m,n) options operator, arrow: (1/6)*m*n*(2*m^2*n^2-m^2-n^2) end proc: for m to 12 do seq(T(m, n), n = 1 .. m) end do; # yields sequence in triangular form

T[m_, n_] := (1/6)*m*n*(2*m^2*n^2 - m^2 - n^2); Table[T[m, n], {m, 1, 12}, {n, 1, m}] // Flatten (* Jean-François Alcover, Feb 09 2018 *)

T(m,n) = mn(2m^2 n^2 - m^2 - n^2)/6. See the Klavzar et al. reference; p. 47, line 6; there is a typo: n^2 - m^2 should be n^2 + m^2.

A100186 Structured heptagonal anti-diamond numbers (vertex structure 7).

Original entry on oeis.org

1, 16, 67, 176, 365, 656, 1071, 1632, 2361, 3280, 4411, 5776, 7397, 9296, 11495, 14016, 16881, 20112, 23731, 27760, 32221, 37136, 42527, 48416, 54825, 61776, 69291, 77392, 86101, 95440, 105431, 116096

Offset: 1

James A. Record (james.record(AT)gmail.com), Nov 07 2004

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A063521 = alternate vertex; A100188 = structured anti-diamonds; A100145 for more on structured numbers.

[(1/6)*(22*n^3-24*n^2+8*n): n in [1..40]]; // Vincenzo Librandi, Aug 18 2011

Table[(22*n^3 - 24*n^2 + 8*n)/6, {n,1,40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 16, 67, 176}, 40] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, (22*n^3 -24*n^2 +8*n)/6) \\ G. C. Greubel, Nov 08 2018

a(n) = (1/6)*(22*n^3 - 24*n^2 + 8*n).
G.f.: x*(1 + 12*x + 9*x^2)/(1-x)^4. - Colin Barker, Jan 19 2012
E.g.f.: (3*x +21*x^2 +11*x^3)*exp(x)/3. - G. C. Greubel, Nov 08 2018

A245827 Szeged index of the grid graph P_3 X P_n.

Original entry on oeis.org

4, 59, 216, 526, 1040, 1809, 2884, 4316, 6156, 8455, 11264, 14634, 18616, 23261, 28620, 34744, 41684, 49491, 58216, 67910, 78624, 90409, 103316, 117396, 132700, 149279, 167184, 186466, 207176, 229365, 253084, 278384, 305316, 333931, 364280, 396414, 430384, 466241, 504036, 543820

Offset: 1

Emeric Deutsch, Aug 06 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
S. Klavzar, A. Rajapakse, I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett., 9, 1996, 45-49.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A245826, A063521, A245828.

[(1/2)*n*(17*n^2 - 9): n in [1..40]]; // Vincenzo Librandi, Aug 07 2014

a := proc (n) options operator, arrow: (1/2)*n*(17*n^2-9) end proc: seq(a(n), n = 1 .. 40);

CoefficientList[Series[(4 x^2 + 43 x + 4)/(x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Aug 07 2014 *)
LinearRecurrence[{4,-6,4,-1},{4,59,216,526},40] (* Harvey P. Dale, Oct 21 2017 *)

Vec(x*(4*x^2+43*x+4)/(x-1)^4 + O(x^100)) \\ Colin Barker, Aug 07 2014

a(n) = (1/2)*n*(17*n^2 - 9).
a(n) = A245826(n, 3).
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). G.f.: x*(4*x^2+43*x+4) / (x-1)^4. - Colin Barker, Aug 07 2014

cp's OEIS Frontend

A004126 a(n) = n*(7*n^2 - 1)/6.

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A063522 a(n) = n*(5*n^2 - 3)/2.

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A004467 a(n) = n*(11*n^2 - 5)/6.

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A062025 a(n) = n*(13*n^2 - 7)/6.

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A063523 a(n) = n*(8*n^2 - 5)/3.

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A100188 Polar structured meta-anti-diamond numbers, the n-th number from a polar structured n-gonal anti-diamond number sequence.

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A166343 Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+12*x+x^2)/(1-x)^4, read by rows.

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A004126 a(n) = n(7n^2 - 1)/6.

A063522 a(n) = n(5n^2 - 3)/2.

A004467 a(n) = n(11n^2 - 5)/6.

A062025 a(n) = n(13n^2 - 7)/6.

A063523 a(n) = n(8n^2 - 5)/3.

A166343 Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)p(n, x)/x, p(n, x) = xD( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x(1+x)/(1-x)^3, and p(3, x) = x(1+12*x+x^2)/(1-x)^4, read by rows.