A004006 -id:A004006 - cp's OEIS frontend

A228074 A Fibonacci-Pascal triangle read by rows: T(n,0) = Fibonacci(n), T(n,n) = n and for n > 0: T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n.

0, 1, 1, 1, 2, 2, 2, 3, 4, 3, 3, 5, 7, 7, 4, 5, 8, 12, 14, 11, 5, 8, 13, 20, 26, 25, 16, 6, 13, 21, 33, 46, 51, 41, 22, 7, 21, 34, 54, 79, 97, 92, 63, 29, 8, 34, 55, 88, 133, 176, 189, 155, 92, 37, 9, 55, 89, 143, 221, 309, 365, 344, 247, 129, 46, 10

Offset: 0

Reinhard Zumkeller, Aug 15 2013

Sum of n-th row is 2^(n+1) - F(n+1) - 1 = A228078(n+1). - Greg Dresden and Sadek Mohammed, Aug 30 2022

			.    0:                                 0
.    1:                               1   1
.    2:                             1   2   2
.    3:                          2    3    4   3
.    4:                       3    5    7    7   4
.    5:                     5    8   12   14   11   5
.    6:                  8   13   20   26   25   16   6
.    7:               13   21   33   46   51   41   22   7
.    8:            21   34   54   79   97   92   63   29   8
.    9:          34   55   88  133  176  189  155   92   37   9
.   10:       55   89  143  221  309  365  344  247  129   46  10
.   11:     89  144  232  364  530  674  709  591  376  175  56   11
.   12:  144 233  376  596  894 1204 1383 1300  967  551  231  67   12 .

Reinhard Zumkeller, Rows n = 0..120 of table, flattened
Index entries for triangles and arrays related to Pascal's triangle

Cf. A000045 (left edge), A001477 (right edge), A228078 (row sums), A027988 (maxima per row);
diagonals T(*,k): A000045, A000071, A001924, A014162, A014166, A053739, A053295, A053296, A053308, A053309;
diagonals T(k,*): A001477, A001245, A004006, A027927, A027928, A027929, A027930, A027931, A027932, A027933;
some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A111006, A114197, A162741.

T:= function(n,k)
    if k=0 then return Fibonacci(n);
    elif k=n then return n;
    else return T(n-1,k-1) + T(n-1,k);
    fi;
  end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Sep 05 2019

a228074 n k = a228074_tabl !! n !! k
a228074_row n = a228074_tabl !! n
a228074_tabl = map fst $ iterate
   (\(u:_, vs) -> (vs, zipWith (+) ([u] ++ vs) (vs ++ [1]))) ([0], [1,1])

with(combinat);
T:= proc (n, k) option remember;
if k = 0 then fibonacci(n)
elif k = n then n
else T(n-1, k-1) + T(n-1, k)
end if
end proc;
seq(seq(T(n, k), k = 0..n), n = 0..12); # G. C. Greubel, Sep 05 2019

T[n_, k_]:= T[n, k]= If[k==0, Fibonacci[n], If[k==n, n, T[n-1, k-1] + T[n -1, k]]]; Table[T[n, k], {n,0,12}, {k,0,n}] (* G. C. Greubel, Sep 05 2019 *)

T(n,k) = if(k==0, fibonacci(n), if(k==n, n, T(n-1, k-1) + T(n-1, k)));
for(n=0, 12, for(k=0, n, print1(T(n,k), ", "))) \\ G. C. Greubel, Sep 05 2019

def T(n, k):
    if (k==0): return fibonacci(n)
    elif (k==n): return n
    else: return T(n-1, k) + T(n-1, k-1)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Sep 05 2019

A004068 Number of atoms in a decahedron with n shells.

Original entry on oeis.org

0, 1, 7, 23, 54, 105, 181, 287, 428, 609, 835, 1111, 1442, 1833, 2289, 2815, 3416, 4097, 4863, 5719, 6670, 7721, 8877, 10143, 11524, 13025, 14651, 16407, 18298, 20329, 22505, 24831, 27312, 29953, 32759, 35735, 38886, 42217, 45733, 49439

Offset: 0

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

Also as a(n)=(n/6)*(5*n^2+1), n>0: structured pentagonal diamond numbers (vertex structure 6) (cf. A081436 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Number of atoms in decahedron with n shells, number = 5/6*(n^3) + 1/6*(n) (T. P. Martin, Shells of atoms, eq.(3)). - Brigitte Stepanov, Jul 02 2011
a(n+1) is the number of triples (w,x,y) having all terms in {0,...,n} and x+y >= w. - Clark Kimberling, Jun 14 2012
a(n) = Sum_{k=1..n} A215630(n,k) for n > 0. - Reinhard Zumkeller, Nov 11 2012
a(n) - a(n-2) = A010001(n-1), for n>1. - K. G. Stier, Dec 21 2012
a(n) is also a figurate number representing a cube of side n with a vertex cut off by a tetrahedron of side n-1. As such, a(n) = A000578(n) - A000292(n-1), n > 0. - Jean M. Morales, Aug 11 2013
The sequence starting with 1 is the third partial sum of (1, 4, 5, 5, 5, ...) and the binomial transform of (1, 6, 10, 5, 0, 0, 0, ...). - Gary W. Adamson, Sep 27 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (3).
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. A000292, A005891, A006322 (partial sums), A010001, A081436, A100145, A215630.

[5*n^3/6+n/6: n in [0..50]]; // Vincenzo Librandi, May 15 2011

Table[5*n^3/6+n/6,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

A004068(n):=5*n^3/6+n/6$ makelist(A004068(n),n,0,20); /* Martin Ettl, Jan 07 2013 */

a(n)=5*n^3/6+n/6 \\ Charles R Greathouse IV, Sep 24 2015

def A004068(n): return n*(5*n**2+1)//6 # Chai Wah Wu, Mar 25 2025

a(n) = 5*binomial(n + 1, 3) + binomial(n, 1).
a(n) = 5*n^3/6 + n/6.
a(n) = Sum_{i=0..n-1} A005891(i). - Xavier Acloque, Oct 08 2003
G.f.: x*(1+3*x+x^2) / (1-x)^4. - R. J. Mathar, Jun 05 2011
E.g.f.: (x/6)*(5x^2 + 15x + 6)*exp(x). - G. C. Greubel, Sep 27 2015
Sum_{n>0} 1/a(n) = 3*(2*gamma + polygamma(0, 1-i/sqrt(5)) + polygamma(0, 1+i/sqrt(5))) = 1.233988011257952852492845364799197179252... where i denotes the imaginary unit. - Stefano Spezia, Aug 31 2023

Typo in definition corrected by Jean M. Morales, Aug 11 2013

A004188 a(n) = n(3n^2 - 1)/2.

Original entry on oeis.org

0, 1, 11, 39, 94, 185, 321, 511, 764, 1089, 1495, 1991, 2586, 3289, 4109, 5055, 6136, 7361, 8739, 10279, 11990, 13881, 15961, 18239, 20724, 23425, 26351, 29511, 32914, 36569, 40485, 44671, 49136, 53889, 58939, 64295, 69966, 75961

Offset: 0

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

3-dimensional analog of centered polygonal numbers.

(1), (4+7), (10+13+16), (19+22+25+28), ... - Jon Perry, Sep 10 2004

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 140.
T. P. Martin, Shells of atoms, Phys. Reports, 273 (1996), 199-241, eq. (11).

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
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. A002412, A016061, A051682.
Cf. A236770 (partial sums).

[n*(3*n^2-1)/2: n in [0..50]]; //Vincenzo Librandi, May 15 2011

seq(binomial(2*n+1,3)+binomial(n+1,3), n=0..37); # Zerinvary Lajos, Jan 21 2007

Table[n(3n^2-1)/2,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)
LinearRecurrence[{4,-6,4,-1},{0,1,11,39},40] (* Harvey P. Dale, Jul 19 2019 *)

PARI
```
vector(40, n, n*(3*n^2-1)/2)
```

Partial sums of n-1 3-spaced triangular numbers, e.g., a(4) = t(1) + t(4) + t(7) = 1 + 10 + 28 = 39. - Jon Perry, Jul 23 2003
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+7*x+x^2) / (x-1)^4. - R. J. Mathar, Oct 08 2011
From Miquel Cerda, Dec 25 2016: (Start)
a(n) = A000578(n) + A135503(n).
a(n) = A007588(n) - A135503(n). (End)
E.g.f.: (x/2)*(2 + 9*x + 3*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A004466 a(n) = n(5n^2 - 2)/3.

Original entry on oeis.org

0, 1, 12, 43, 104, 205, 356, 567, 848, 1209, 1660, 2211, 2872, 3653, 4564, 5615, 6816, 8177, 9708, 11419, 13320, 15421, 17732, 20263, 23024, 26025, 29276, 32787, 36568, 40629, 44980, 49631, 54592

Offset: 0

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

3-dimensional analog of centered polygonal numbers.
Also as a(n)=(1/6)*(10*n^3-4*n), n>0: structured pentagonal anti-diamond numbers (vertex structure 11) (Cf. A051673 = alternate vertex A100188 = structured anti-diamonds; A100145 for more on structured numbers). - James A. Record (james.record(AT)gmail.com), Nov 07 2004
a(n+1)-10*a(n) = (n+1)*(5*(n+1)^2-2)/3 - (10n(n+1)(n+2)/6) = n. The unit digits are 0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,... . - Eric Desbiaux, Aug 18 2008

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).

Cf. A062786 (first differences), A264853 (partial sums).

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*(5*n^2-2)/3: n in [0..50]]; // Vincenzo Librandi, May 15 2011

A004466:=n->n*(5*n^2 - 2)/3; seq(A004466(n), n=0..50); # Wesley Ivan Hurt, Mar 10 2014

Table[n(5n^2-2)/3,{n,0,80}] (* Vladimir Joseph Stephan Orlovsky, Apr 18 2011 *)

a(n)=n*(5*n^2-2)/3 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x*(1+8*x+x^2)/(1-x)^4. - Colin Barker, Jan 08 2012

E.g.f.: (x/3)*(3 + 15*x + 5*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

A063521 a(n) = n(7n^2-4)/3.

Original entry on oeis.org

0, 1, 16, 59, 144, 285, 496, 791, 1184, 1689, 2320, 3091, 4016, 5109, 6384, 7855, 9536, 11441, 13584, 15979, 18640, 21581, 24816, 28359, 32224, 36425, 40976, 45891, 51184, 56869, 62960, 69471, 76416, 83809, 91664, 99995, 108816, 118141, 127984, 138359, 149280, 160761

Offset: 0

N. J. A. Sloane, Aug 02 2001

Also as a(n)=(1/6)*(14*n^3-8*n), n>0: structured heptagonal anti-diamond numbers (vertex structure 15) (Cf. A100186 = 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.

A063521:=n->n*(7*n^2-4)/3; seq(A063521(k), k=0..100); # Wesley Ivan Hurt, Oct 24 2013

lst={};Do[AppendTo[lst, n*(7*n^2-4)/3], {n, 1, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 02 2008 *)
CoefficientList[Series[x*(1+12*x+x^2)/(1-x)^4, {x, 0, 50}], x] (* G. C. Greubel, Sep 01 2017 *)

a(n) = { n*(7*n^2 - 4)/3 } \\ Harry J. Smith, Aug 25 2009

G.f.: x*(1+12*x+x^2)/(1-x)^4. - Colin Barker, Jan 10 2012

E.g.f.: (x/3)*(3 + 21*x + 7*x^2)*exp(x). - G. C. Greubel, Sep 01 2017

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

Original entry on oeis.org

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

A006522 4-dimensional analog of centered polygonal numbers. Also number of regions created by sides and diagonals of a convex n-gon in general position.

Original entry on oeis.org

1, 0, 0, 1, 4, 11, 25, 50, 91, 154, 246, 375, 550, 781, 1079, 1456, 1925, 2500, 3196, 4029, 5016, 6175, 7525, 9086, 10879, 12926, 15250, 17875, 20826, 24129, 27811, 31900, 36425, 41416, 46904, 52921, 59500, 66675, 74481, 82954, 92131

Offset: 0

N. J. A. Sloane

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=5, a(n)=coeff(charpoly(A,x),x^(n-4)). - Milan Janjic, Jan 24 2010
From Ant King, Sep 14 2011: (Start)
Consider the array formed by the polygonal numbers of increasing rank A139600
  0, 1, 3,  6, 10, 15, 21,  28,  36,  45, ...  A000217(n)
  0, 1, 4,  9, 16, 25, 36,  49,  64,  81, ...  A000290(n)
  0, 1, 5, 12, 22, 35, 51,  70,  92, 117, ...  A000326(n)
  0, 1, 6, 15, 28, 45, 66,  91, 120, 153, ...  A000384(n)
  0, 1, 7, 18, 34, 55, 81, 112, 148, 189, ...  A000566(n)
  0, 1, 8, 21, 40, 65, 96, 133, 176, 225, ...  A000567(n)
  ...
Then, for n>=2, a(n) is the diagonal sum of this polygonal grid.
(End)
Binomial transform of (1, -1, 1, 0, 1, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

			For a pentagon in general position, 11 regions are formed (Comtet, Fig. 20, p. 74).

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.
Ross Honsberger, Mathematical Gems, M.A.A., 1973, p. 102.
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..1000
J. W. Freeman, The number of regions determined by a convex polygon, Math. Mag., 49 (1976), 23-25.
Math Forum, Regions of a circle Cut by Chords to n points.
V. Meally, Letter to N. J. A. Sloane, May 1975.
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
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.
Eric Weisstein's World of Mathematics, Polygon Diagonal.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Partial sums of A004006.

Cf. A000332, A000217.

[Binomial(n, 4)+Binomial(n-1, 2): n in [0..40]]; // Vincenzo Librandi, Jun 09 2013

A006522 := n->(1/24)*(n-1)*(n-2)*(n^2-3*n+12):
seq(A006522(n), n=0..40);
A006522:=-(1-z+z**2)/(z-1)**5; # Simon Plouffe in his 1992 dissertation; gives sequence except for three leading terms

a=2;b=3;s=4;lst={1,0,0,1,s};Do[a+=n;b+=a;s+=b;AppendTo[lst,s],{n,2,6!,1}];lst (* Vladimir Joseph Stephan Orlovsky, May 24 2009 *)
Table[Binomial[n,4]+Binomial[n-1,2],{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,0,0,1,4},40] (* Harvey P. Dale, Jul 11 2011 *)
CoefficientList[Series[-(((x - 1) x (x (4 x - 5) + 5) + 1) / (x - 1)^5), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)
a[n_] := If[n==0, 1, Sum[PolygonalNumber[n-k+1, k], {k, 0, n-2}]];
a /@ Range[0, 40] (* Jean-François Alcover, Jan 21 2020 *)

a(n)=1/24*n^4 - 1/4*n^3 + 23/24*n^2 - 7/4*n + 1 \\ Charles R Greathouse IV, Feb 09 2017

a(n) = binomial(n,4) + binomial(n-1,2) = A000332(n) + A000217(n-2).
a(n) = binomial(n-1,2) + binomial(n-1,3) + binomial(n-1,4). - Zerinvary Lajos, Jul 23 2006
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=1, a(1)=0, a(2)=0, a(3)=1, a(4)=4. - Harvey P. Dale, Jul 11 2011
G.f.: -((x-1)*x*(x*(4*x-5)+5)+1)/(x-1)^5. - Harvey P. Dale, Jul 11 2011
a(n) = (n^4 - 6*n^3 + 23*n^2 - 42*n + 24)/24. - T. D. Noe, Oct 16 2013
For odd n, a(n) = A007678(n). - R. J. Mathar, Nov 22 2017
a(n) = a(3-n) for all n in Z. - Michael Somos, Nov 23 2021
Sum_{n>=3} 1/a(n) = 66/25 - (4/5)*sqrt(3/13)*Pi*tanh(sqrt(39)*Pi/2). - Amiram Eldar, Aug 23 2022

A292508 Number A(n,k) of partitions of n with k kinds of 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 1, 1, 3, 4, 3, 2, 1, 4, 7, 7, 5, 2, 1, 5, 11, 14, 12, 7, 4, 1, 6, 16, 25, 26, 19, 11, 4, 1, 7, 22, 41, 51, 45, 30, 15, 7, 1, 8, 29, 63, 92, 96, 75, 45, 22, 8, 1, 9, 37, 92, 155, 188, 171, 120, 67, 30, 12, 1, 10, 46, 129, 247, 343, 359, 291, 187, 97, 42, 14

Offset: 0

Alois P. Heinz, Sep 17 2017

Partial sum operator applied to column k gives column k+1.

A(n,k) is also defined for k < 0. All given formulas and programs can be applied also if k is negative.

			Square array A(n,k) begins:
  1,  1,  1,   1,   1,    1,    1,    1,     1, ...
  0,  1,  2,   3,   4,    5,    6,    7,     8, ...
  1,  2,  4,   7,  11,   16,   22,   29,    37, ...
  1,  3,  7,  14,  25,   41,   63,   92,   129, ...
  2,  5, 12,  26,  51,   92,  155,  247,   376, ...
  2,  7, 19,  45,  96,  188,  343,  590,   966, ...
  4, 11, 30,  75, 171,  359,  702, 1292,  2258, ...
  4, 15, 45, 120, 291,  650, 1352, 2644,  4902, ...
  7, 22, 67, 187, 478, 1128, 2480, 5124, 10026, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A002865, A000041, A000070, A014153, A014160, A014161, A120477, A320753, A320754, A320755, A320756.
Rows n=0-4 give: A000012, A001477, A000124, A004006(k+1), A027927(k+3).
Main diagonal gives A292463.
A(n,n+1) gives A292613.
Cf. A292622, A292741, A292745.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      (numtheory[sigma](j)+k-1)*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1, `if`(k<1,
      A(n, k+1)-A(n-1, k+1), `if`(k=1, combinat[numbpart](n),
      A(n-1, k)+A(n, k-1))))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
# third Maple program:
b:= proc(n, i, k) option remember; `if`(n=0 or i<2,
      binomial(k+n-1, n), add(b(n-i*j, i-1, k), j=0..n/i))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i < 2, Binomial[k + n - 1, n], Sum[b[n - i*j, i - 1, k], {j, 0, n/i}]];
A[n_, k_] := b[n, n, k];
Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 17 2018, translated from 3rd Maple program *)

G.f. of column k: 1/(1-x)^k * 1/Product_{j>1} (1-x^j).
Column k is Euler transform of k,1,1,1,... .
For fixed k>=0, A(n,k) ~ 2^((k-5)/2) * 3^((k-2)/2) * n^((k-3)/2) * exp(Pi*sqrt(2*n/3)) / Pi^(k-1). - Vaclav Kotesovec, Oct 24 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

cp's OEIS Frontend

A228074 A Fibonacci-Pascal triangle read by rows: T(n,0) = Fibonacci(n), T(n,n) = n and for n > 0: T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n.

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A004068 Number of atoms in a decahedron with n shells.

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A004188 a(n) = n*(3*n^2 - 1)/2.

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A004466 a(n) = n*(5*n^2 - 2)/3.

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A063521 a(n) = n*(7*n^2-4)/3.

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

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A004188 a(n) = n(3n^2 - 1)/2.

A004466 a(n) = n(5n^2 - 2)/3.

A063521 a(n) = n(7n^2-4)/3.

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

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

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