A000579 -id:A000579 - cp's OEIS frontend

A103252 Array A000292(n)*A000217(k), read by antidiagonals.

1, 4, 3, 10, 12, 6, 20, 30, 24, 10, 35, 60, 60, 40, 15, 56, 105, 120, 100, 60, 21, 84, 168, 210, 200, 150, 84, 28, 120, 252, 336, 350, 300, 210, 112, 36, 165, 360, 504, 560, 525, 420, 280, 144, 45, 220, 495, 720, 840, 840, 735, 560, 360, 180, 55

Offset: 1

Gary W. Adamson, Mar 20 2005

			Array begins
   1,   3,   6,  10,  15,  21,  28,   36,   45,   55, ...
   4,  12,  24,  40,  60,  84, 112,  144,  180,  220, ...
  10,  30,  60, 100, 150, 210, 280,  360,  450,  550, ...
  20,  60, 120, 200, 300, 420, 560,  720,  900, 1100, ...
  35, 105, 210, 350, 525, 735, 980, 1260, 1575, 1925, ...
  ...

Isabel Cação, Helmuth R. Malonek, Maria Irene Falcão, and Graça Tomaz, Intrinsic Properties of a Non-Symmetric Number Triangle, J. Int. Seq., Vol. 26 (2023), Article 23.4.8.
Joaquín Figueroa, Ivan Gonzalez, and Daniel Salinas-Arizmendi, A Novel Transfer Matrix Framework for Multiple Dirac Delta Potentials, arXiv:2503.23134 [quant-ph], 2025. See pp. 5, 9.

Cf. A000579 (antidiagonal sums).
Main diagonal gives A004302.
Cf. A000217, A000292.

A[n_,k_]:=Binomial[n+2,3]Binomial[k+1,2]; Table[A[n-k+1,k],{n,10},{k,n}]//Flatten (* Stefano Spezia, May 21 2023 *)

G.f.: x*y/((1 - x)^4*(1 - y)^3). - Stefano Spezia, May 21 2023

More terms from Stefano Spezia, May 21 2023

A116690 a(n) = C(n,8) + C(n,7) + C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).

Original entry on oeis.org

0, 1, 3, 7, 15, 31, 63, 127, 255, 510, 1012, 1980, 3796, 7098, 12910, 22818, 39202, 65535, 106761, 169765, 263949, 401929, 600369, 880969, 1271625, 1807780, 2533986, 3505698, 4791322, 6474540, 8656936, 11460948, 15033172, 19548045

Offset: 0

Jonathan Vos Post, Mar 15 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

[n*(n+1)*(25584 - 9604*n + 5264*n^2 - 1295*n^3 + 231*n^4 - 21*n^5 + n^6 ) /40320: n in [0..30]]; // G. C. Greubel, Nov 25 2017

seq(sum(binomial(n,k),k=1..8),n=0..33); # Zerinvary Lajos, Dec 14 2007

Table[n*(n + 1)*(25584 - 9604*n + 5264*n^2 - 1295*n^3 + 231*n^4 - 21*n^5 + n^6)/40320, {n, 0, 50}] (* G. C. Greubel, Nov 25 2017 *)
Table[Total[Binomial[n,Range[8]]],{n,0,40}] (* Harvey P. Dale, Aug 14 2023 *)

for(n=0,30, print1(n*(n+1)*(25584 - 9604*n + 5264*n^2 - 1295*n^3 + 231*n^4 - 21*n^5 + n^6 ) /40320, ", ")) \\ G. C. Greubel, Nov 25 2017

[binomial(n,2)+binomial(n,4)+binomial(n,6)+binomial(n,8) for n in range(1, 35)] # Zerinvary Lajos, May 17 2009

[binomial(n,2)+binomial(n,4)+binomial(n,6)+binomial(n,8)+binomial(n,1)+binomial(n,3)+binomial(n,5)+binomial(n,7)for n in range(0, 34)] # Zerinvary Lajos, May 17 2009

a(n) = A000581(n) + A000580(n) + A000579(n) + A000389(n) + A000332(n) + A000292(n) + A000217(n) + n. a(n) = A000581(n) + A116082(n).
G.f. ( -x*(2*x^2 - 2*x + 1)*(2*x^4 - 4*x^3 + 6*x^2 - 4*x + 1) ) / (x-1)^9. - R. J. Mathar, Oct 21 2011
a(n) = n*(n+1)*(25584 - 9604*n + 5264*n^2 - 1295*n^3 + 231*n^4 - 21*n^5 + n^6)/40320. - G. C. Greubel, Nov 25 2017

A122960 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 1, 0, 6, 1, 0, 0, 5, 0, 10, 1, 0, 1, 0, 15, 0, 15, 1, 0, 0, 7, 0, 35, 0, 21, 1, 0, 1, 0, 28, 0, 70, 0, 28, 1, 0, 0, 9, 0, 84, 0, 126, 0, 36, 1, 0, 1, 0, 45, 0, 210, 0, 210, 0, 45, 1

Offset: 0

Philippe Deléham, Oct 26 2006

T(n,k) = binomial (n,n-k+1) if (n-k) is an odd number (see A000217, A000332, A000579, A000581, ...). T(n,k)= 0 if (n-k)=2x with x > 0 (see A000004). T(n,n)=1 (see A000012).

			Triangle begins:
  1;
  0, 1;
  0, 1, 1;
  0, 0, 3,  1;
  0, 1, 0,  6,  1;
  0, 0, 5,  0, 10,   1;
  0, 1, 0, 15,  0,  15,   1;
  0, 0, 7,  0, 35,   0,  21,   1;
  0, 1, 0, 28,  0,  70,   0,  28,  1;
  0, 0, 9,  0, 84,   0, 126,   0, 36,  1;
  0, 1, 0, 45,  0, 210,   0, 210,  0, 45, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A000004, A000012, A000217, A000332, A000579, A000581, A007318.

T:= proc(n, k) option remember;
      if k<0 or k>n then 0
    elif k=n then 1
    elif n=2 and k=1 then 1
    elif k=0 then 0
    else 3*T(n-1, k-1) + T(n-2, k) - 3*T(n-2, k-2) - T(n-3, k-1) + T(n-3, k-3)
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Feb 17 2020

With[{m = 10}, CoefficientList[CoefficientList[Series[(1-2*x*y-x^2+x^2*y^2+
x^2*y)/(1-3*x*y-x^2+3*x^2*y^2+x^3*y-x^3*y^3), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)

T(n, k) = if(k<0 || k>n, 0, if(k==n, 1, if(n==2 && k==1, 1, if(k==0, 0, 3*T(n-1, k-1) + T(n-2, k) - 3*T(n-2, k-2) - T(n-3, k-1) + T(n-3, k-3) )))); \\ G. C. Greubel, Feb 17 2020

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif (k==n): return 1
    elif (n==2 and k==1): return 1
    elif (k==0): return 0
    else: return 3*T(n-1, k-1) + T(n-2, k) - 3*T(n-2, k-2) - T(n-3, k-1) + T(n-3, k-3)
print([[T(n, k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 17 2020

Sum_{k=0..n} T(n,k) = A011782(n).
Sum_{k=0..n} 2^k*T(n,k) = A083323(n).
Sum_{k=0..n} 2^(n-k)*T(n,k) = A122983(n).
G.f.: (1 - 2*x*y - x^2 + x^2*y^2 + x^2*y)/(1 - 3*x*y - x^2 + 3*x^2*y^2 + x^3*y - x^3*y^3). - Philippe Deléham, Nov 09 2013
T(n,k) = 3*T(n-1,k-1) + T(n-2,k) - 3*T(n-2,k-2) - T(n-3,k-1) + T(n-3,k-3), T(0,0) = T(1,1) = T(2,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 09 2013

a(12) corrected by Georg Fischer, Feb 17 2020

A176566 Triangle T(n, k) = binomial(n*(n+1)/2 + k, k), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 4, 10, 20, 1, 7, 28, 84, 210, 1, 11, 66, 286, 1001, 3003, 1, 16, 136, 816, 3876, 15504, 54264, 1, 22, 253, 2024, 12650, 65780, 296010, 1184040, 1, 29, 435, 4495, 35960, 237336, 1344904, 6724520, 30260340, 1, 37, 703, 9139, 91390, 749398, 5245786, 32224114, 177232627, 886163135

Offset: 0

Roger L. Bagula, Apr 20 2010

			Square array of T(n, k):
  1,  1,   1,    1,     1,     1,      1 ...
  1,  1,   1,    1,     1,     1,      1 ... A000012;
  1,  2,   3,    4,     5,     6,      7 ... A000027;
  1,  4,  10,   20,    35,    56,     84 ... A000292;
  1,  7,  28,   84,   210,   462,    924 ... A000579;
  1, 11,  66,  286,  1001,  3003,   8008 ... A001287;
  1, 16, 136,  816,  3876, 15504,  54264 ... A010968;
  1, 22, 253, 2024, 12650, 65780, 296010 ... A010974;
Triangle begins as:
  1;
  1,  1;
  1,  2,   3;
  1,  4,  10,   20;
  1,  7,  28,   84,   210;
  1, 11,  66,  286,  1001,   3003;
  1, 16, 136,  816,  3876,  15504,   54264;
  1, 22, 253, 2024, 12650,  65780,  296010,  1184040;
  1, 29, 435, 4495, 35960, 237336, 1344904,  6724520,  30260340;
  1, 37, 703, 9139, 91390, 749398, 5245786, 32224114, 177232627, 886163135;

G. C. Greubel, Rows n = 0..50 of the triangle, flatten

Cf. A107868 (rows sums), A158498.

[Binomial(Binomial(n, 2) + k, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 09 2021

T[n_, k_]= Binomial[Binomial[n, 2] + k, k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

row(n) = vector(n+1, k, k--; binomial(binomial(n,2) + k, k)); \\ Michel Marcus, Jul 10 2021

flatten([[binomial(binomial(n,2) +k, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 09 2021

T(n, k) = binomial(binomial(n, 2) + k, k).

Sum_{k=0..n} T(n, k) = A107868(n).

A256860 a(n) = n(n + 1)(n + 2)(n + 3)(n^2 - n + 5)/120.

Original entry on oeis.org

1, 7, 33, 119, 350, 882, 1974, 4026, 7623, 13585, 23023, 37401, 58604, 89012, 131580, 189924, 268413, 372267, 507661, 681835, 903210, 1181510, 1527890, 1955070, 2477475, 3111381, 3875067, 4788973, 5875864, 7161000, 8672312, 10440584, 12499641, 14886543

Offset: 1

Luciano Ancora, Apr 14 2015

This is the case k = n of b(n,k) = n*(n+1)*(n+2)*(n+3)*(k*(n-1)+5)/120, where b(n,k) is the n-th hypersolid number in 5 dimensions generated from an arithmetical progression with the first term 1 and common difference k (see Sardelis et al. paper).

D. A. Sardelis and T. M. Valahas, On Multidimensional Pythagorean Numbers, arXiv:0805.4070v1 [math.GM], 2008.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000579.

Cf. similar sequences listed in A256859.

Table[n (n + 1) (n + 2) (n + 3) (n^2 - n + 5)/120, {n, 40}]

vector(40, n, n*(n+1)*(n+2)*(n+3)*(n^2-n+5)/120) \\ Bruno Berselli, Apr 15 2015

G.f.: x*(1 + 5*x^2)/(1 - x)^7.

a(n) = 5*A000579(n+3) + A000579(n+5). [Bruno Berselli, Apr 15 2015]

A260161 a(n) is the a(n-1)-th a(n-2)-dimensional simplex number, starting with the terms 1, 2.

Original entry on oeis.org

1, 2, 2, 3, 6, 56, 55525372

Offset: 1

Robin Powell, Nov 09 2015

nonn

The next term is too large to include.

			56 is the 6th 3-simplex (or tetrahedral) number = A000292(6).
55525372 is the 56th 6-simplex number = A000579(61).
The next term will be the 55525372th 56-simplex number.

OEIS Wiki, Simplicial polytopic numbers

Cf. A000217, A000292, A000579.

A266387 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 322560.

Original entry on oeis.org

0, 0, 0, 0, 0, 7, 42, 147, 392, 882, 1764, 3234, 5544, 9009, 14014, 21021, 30576, 43316, 59976, 81396, 108528, 142443, 184338, 235543, 297528, 371910, 460460, 565110, 687960, 831285, 997542, 1189377, 1409632, 1661352, 1947792, 2272424, 2638944, 3051279

Offset: 1

Philippe A.J.G. Chevalier, Dec 28 2015

The sequence was discovered by enumerating all orbits of Aut(Z^7) and sorting the orbits as function of the infinity norm of the representative integer lattice points. This sequence is one of the 30 sequences that are obtained by classifying the orbits in a table with the rows being the infinity norm and the columns being the 30 cardinalities (1, 14, 84, 128, 168, 280, 448, 560, 672, 840, 896, 1680, 2240, 2688, 3360, 4480, 5376, 6720, 8960, 13440, 17920, 20160, 26880, 40320, 53760, 80640, 107520, 161280, 322560, 645120) generated by signed permutations of integer lattice points of Z^7.

The continued fraction expansion of this sequence is finite and simplifies to the g.f. 7*x^6/(1-x)^6 (see Mathematica). - Benedict W. J. Irwin, Feb 09 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Other sequences that give the number of orbits of Aut(Z^7) as function of the infinity norm for different cardinalities of these orbits: A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

Join[{0, 0, 0, 0, 0},Table[Abs[SeriesCoefficient[Series[7/(x+6/(x - 5/2/(x + ContinuedFractionK[If[Mod[k, 2] ==0, (7 + k/2)/(6 + 2 k), ((k + 1)/2 - 5)/(2 (k - 1) +6)], x, {k, 0, 8}]))), {x, Infinity, 101}],2 n + 1]], {n, 0, 50}]] - (* Benedict W. J. Irwin, Feb 09 2016 *)

concat(vector(5), Vec(7*x^6/(1-x)^6 + O(x^50))) \\ Colin Barker, May 04 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = 7*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)/120.
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) for n>6.
G.f.: 7*x^6 / (1-x)^6.
(End)

A266395 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 161280.

Original entry on oeis.org

0, 0, 0, 0, 15, 75, 225, 525, 1050, 1890, 3150, 4950, 7425, 10725, 15015, 20475, 27300, 35700, 45900, 58140, 72675, 89775, 109725, 132825, 159390, 189750, 224250, 263250, 307125, 356265, 411075, 471975, 539400, 613800, 695640, 785400, 883575, 990675, 1107225

Offset: 1

Philippe A.J.G. Chevalier, Dec 29 2015

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

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

concat(vector(4), Vec(15*x^5/(1-x)^5 + O(x^50))) \\ Colin Barker, May 05 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = 5*(n-1)*(n-2)*(n-3)*(n-4)/8 = 15*A000332(n-1).
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>5.
G.f.: 15*x^5 / (1-x)^5.
(End)

A266396 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 80640.

Original entry on oeis.org

0, 0, 0, 10, 41, 105, 215, 385, 630, 966, 1410, 1980, 2695, 3575, 4641, 5915, 7420, 9180, 11220, 13566, 16245, 19285, 22715, 26565, 30866, 35650, 40950, 46800, 53235, 60291, 68005, 76415, 85560, 95480, 106216, 117810, 130305, 143745, 158175, 173641, 190190

Offset: 1

Philippe A.J.G. Chevalier, Dec 29 2015

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

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

LinearRecurrence[{5,-10,10,-5,1},{0,0,0,10,41},50] (* Harvey P. Dale, Nov 18 2024 *)

concat(vector(3), Vec(x^4*(10-9*x)/(1-x)^5 + O(x^50))) \\ Colin Barker, May 05 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = (n^4+30*n^3-205*n^2+390*n-216)/24.
a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5) for n>5.
G.f.: x^4*(10-9*x) / (1-x)^5.
(End)

A266397 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 26880.

Original entry on oeis.org

0, 0, 9, 31, 70, 130, 215, 329, 476, 660, 885, 1155, 1474, 1846, 2275, 2765, 3320, 3944, 4641, 5415, 6270, 7210, 8239, 9361, 10580, 11900, 13325, 14859, 16506, 18270, 20155, 22165, 24304, 26576, 28985, 31535, 34230, 37074, 40071, 43225, 46540, 50020, 53669

Offset: 1

Philippe A.J.G. Chevalier, Dec 29 2015

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

Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A001477, A000217.

concat(vector(2), Vec(x^3*(9-5*x)/(1-x)^4 + O(x^50))) \\ Colin Barker, May 05 2016

From Colin Barker, Dec 29 2015: (Start)
a(n) = (4*n^3+3*n^2-37*n+30)/6.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.
G.f.: x^3*(9-5*x) / (1-x)^4.
(End)

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A103252 Array A000292(n)*A000217(k), read by antidiagonals.

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A116690 a(n) = C(n,8) + C(n,7) + C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).

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A122960 Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1, -1, -1, 1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

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A176566 Triangle T(n, k) = binomial(n*(n+1)/2 + k, k), read by rows.

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A256860 a(n) = n*(n + 1)*(n + 2)*(n + 3)*(n^2 - n + 5)/120.

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A260161 a(n) is the a(n-1)-th a(n-2)-dimensional simplex number, starting with the terms 1, 2.

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A266387 Number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 322560.

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A256860 a(n) = n(n + 1)(n + 2)(n + 3)(n^2 - n + 5)/120.