A019298 -id:A019298 - cp's OEIS frontend

A236762 Number of partitions of 3n into 3 parts with the middle part prime.

0, 2, 5, 7, 11, 14, 17, 19, 23, 29, 35, 40, 47, 53, 59, 67, 76, 82, 88, 93, 100, 109, 118, 124, 131, 140, 149, 160, 173, 185, 197, 208, 220, 232, 244, 258, 273, 285, 297, 311, 327, 342, 357, 369, 382, 397, 412, 426, 442, 460, 478, 496, 515, 533, 551, 571

Offset: 1

Wesley Ivan Hurt, Jan 30 2014

			Count the primes in the second columns for a(n):
                                               13 + 1 + 1
                                               12 + 2 + 1
                                               11 + 3 + 1
                                               10 + 4 + 1
                                                9 + 5 + 1
                                                8 + 6 + 1
                                                7 + 7 + 1
                                   10 + 1 + 1  11 + 2 + 2
                                    9 + 2 + 1  10 + 3 + 2
                                    8 + 3 + 1   9 + 4 + 2
                                    7 + 4 + 1   8 + 5 + 2
                                    6 + 5 + 1   7 + 6 + 2
                        7 + 1 + 1   8 + 2 + 2   9 + 3 + 3
                        6 + 2 + 1   7 + 3 + 2   8 + 4 + 3
                        5 + 3 + 1   6 + 4 + 2   7 + 5 + 3
                        4 + 4 + 1   5 + 5 + 2   6 + 6 + 3
            4 + 1 + 1   5 + 2 + 2   6 + 3 + 3   7 + 4 + 4
            3 + 2 + 1   4 + 3 + 2   5 + 4 + 3   6 + 5 + 4
1 + 1 + 1   2 + 2 + 2   3 + 3 + 3   4 + 4 + 4   5 + 5 + 5
   3(1)        3(2)        3(3)        3(4)        3(5)     ..   3n
--------------------------------------------------------------------
    0           2           5           7          11      ..   a(n)

Index entries for sequences related to partitions

Cf. A019298, A235988, A236364, A236758, A010051.

with(numtheory); A236762:=n->sum( i * (pi(i) - pi(i - 1)), i = 1..n) +
sum( (pi(n + i) - pi(n + i - 1)) * (n - 2*i), i = 1..floor((n - 1)/2) ); seq(A236762(n), n=1..100);

Table[Sum[i (PrimePi[i] - PrimePi[i - 1]), {i, n}] + Sum[(PrimePi[n + i] - PrimePi[n + i - 1]) (n - 2 i), {i, Floor[(n - 1)/2]}], {n, 100}]

def a(n): return sum(1 for L in Partitions(3*n,length=3).list() if is_prime(L[1])) # Ralf Stephan, Feb 03 2014

a(n) = Sum_{i=1..n} i * A010051(i) + Sum_{i=1..floor((n - 1)/2)} A010051(n + i) * (n - 2i).

A293311 Rectangular array read by antidiagonals: A(n,k) = number of magic labelings of the graph LOOP X C_n (see comments) having magic sum k, n >= 1, k >= 0.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 11, 7, 1, 6, 15, 23, 26, 11, 1, 7, 21, 42, 70, 57, 18, 1, 8, 28, 69, 155, 197, 129, 29, 1, 9, 36, 106, 301, 533, 571, 289, 47, 1, 10, 45, 154, 532, 1223, 1884, 1640, 650, 76, 1, 11, 55, 215, 876, 2494, 5103, 6604, 4726, 1460, 123, 1

Offset: 1

L. Edson Jeffery, Oct 06 2017

The graph LOOP X C_n is constructed by attaching a loop to each vertex of the cycle graph C_n.

The generating function for row n of this array was found via the "Omega" package for Mathematica authored by Axel Riese. The package can be downloaded from the link given in the article by G. E. Andrews, et al., and is needed for the Mathematica program below.

			Array begins:
.  1   2    3     4      5       6       7        8        9        10
.  1   3    6    10     15      21      28       36       45        55
.  1   4   11    23     42      69     106      154      215       290
.  1   7   26    70    155     301     532      876     1365      2035
.  1  11   57   197    533    1223    2494     4654     8105     13355
.  1  18  129   571   1884    5103   11998    25362    49347     89848
.  1  29  289  1640   6604   21122   57271   137155   298184    599954
.  1  47  650  4726  23219   87677  274132   743724  1806597   4016683
.  1  76 1460 13604  81555  363606 1310974  4029310 10936124  26868719
.  1 123 3281 39175 286555 1508401 6271378 21836366 66220705 179784715

G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package.
Eric Weisstein's World of Mathematics, Cycle Graph.
Eric Weisstein's World of Mathematics, Graph Loop.

Cf. A293311.

Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310, A293309 (rows 1..10).

(* Run this first: *)
<< Omega.m;
(* Then run the following in a different cell: *)
nmax = 11; Do[cond = {}; Do[If[n == 1, AppendTo[cond, Subscript[a, 1] + Subscript[a, 2] == Subscript[a, 3]]; Break[], AppendTo[cond, If[j == n, Subscript[a, 2*j - 1] + Subscript[a, 2*j] + Subscript[a, 1] == Subscript[a, 2*n + 1], Subscript[a, 2*j - 1] + Subscript[a, 2*j] + Subscript[a, 2*j + 1] == Subscript[a, 2*n + 1]]]], {j, n}]; f = OEqSum[Product[Subscript[x, i]^Subscript[a, i], {i, 2*n + 1}], cond, \[Lambda]][[1]] /. {Subscript[x, 2*n + 1] -> z} /. {Subscript[x, _] -> 1}; Do[f = OEqR[f, Subscript[\[Lambda], k]], {k, Length[cond]}]; Do[a293311[n, k] = Coefficient[Series[f, {z, 0, nmax - 1}], z, k], {k, 0, nmax - 1}], {n, nmax}];
(* Array: *)
Grid[Table[a293311[n, k], {n, nmax}, {k, 0, nmax - 1}]]
(* Array antidiagonals flattened (gives this sequence): *)
Flatten[Table[a293311[n, k - n], {k, 11}, {n, k}]]

A003438 Number of 5 X 5 matrices with nonnegative integer entries and row and column sums equal to n.

Original entry on oeis.org

1, 120, 6210, 153040, 2224955, 22069251, 164176640, 976395820, 4855258305, 20856798285, 79315936751, 272095118010, 854560160105, 2486299719645, 6765755480415, 17356306529251, 42250330784180, 98137852369965

Offset: 0

N. J. A. Sloane

Number of 5 X 5 stochastic matrices of integers.

D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, p. 234.

T. D. Noe, Table of n, a(n) for n = 0..1000
D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Cf. A002817, A001496, A019298.
Cf. A001496, A005466, A058391.
Row n=5 of A257493.

CoefficientList[Series[(1+103x+4306x^2+63110x^3+388615x^4+1115068x^5+ 1575669x^6+1115068x^7+388615x^8+63110x^9+4306x^10+103x^11+x^12)/ (1-x)^17,{x,0,30}],x] (* Harvey P. Dale, Aug 17 2013 *)

G.f.: (1 + 103*x + 4306*x^2 + 63110*x^3 + 388615*x^4 + 1115068*x^5 + 1575669*x^6 + 1115068*x^7 + 388615*x^8 + 63110*x^9 + 4306*x^10 + 103*x^11 + x^12)/(1-x)^17.

a(n) = Sum_{j=0..6} A005466(j) * binomial(4+j+n, 4+2*j). - Andrew Howroyd, Apr 09 2020

More terms from Vladeta Jovovic, Feb 06 2000

A220075 Partial sums in rows of A220073, triangle read by rows.

Original entry on oeis.org

1, 1, 3, 2, 3, 6, 3, 4, 6, 10, 4, 6, 7, 10, 15, 5, 8, 9, 11, 15, 21, 6, 10, 12, 13, 16, 21, 28, 7, 12, 15, 16, 18, 22, 28, 36, 8, 14, 18, 20, 21, 24, 29, 36, 45, 9, 16, 21, 24, 25, 27, 31, 37, 45, 55, 10, 18, 24, 28, 30, 31, 34, 39, 46, 55, 66, 11, 20, 27

Offset: 1

Reinhard Zumkeller, Dec 03 2012

T(n,k) = sum(A220073(n,i): i=1..k).

Reinhard Zumkeller, Table of n, a(n) for n = 1..7260

Cf. A000027 (left edge), A000217 (right edge), A002061 (central terms), A019298 (row sums); A220053.

a220075 n k = a220075_tabl !! (n-1) !! (k-1)
a220075_row n = a220075_tabl !! (n-1)
a220075_tabl = map (scanl1 (+)) a220073_tabl

A[n_, k_] := If[k == 1, n, If[k == n, n-1, Abs[2k-n-If[2k <= n+1, 2, 1]]]];
A220073[n_, k_] := A[n, n-k+1];
T[n_, k_] := Sum[A220073[n, i], {i, 1, k}];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 27 2021 *)

A293309 Number of magic labelings of the graph LOOP X C_10 (see comments) having magic sum n, n >= 0.

Original entry on oeis.org

1, 123, 3281, 39175, 286555, 1508401, 6271378, 21836366, 66220705, 179784715, 445824731, 1025102013, 2211041131, 4514532465, 8789910980, 16416797116, 29556115153, 51502789451, 87162399205, 143684487475, 231291309931, 364347612673, 562724586326

Offset: 0

L. Edson Jeffery, Oct 05 2017

nonn

The graph LOOP X C_n is constructed by attaching a loop to each vertex of the cycle graph C_n.

The generating function for this sequence was found via the "Omega" package for Mathematica authored by Axel Riese. The package can be downloaded from the link given in the article by G. E. Andrews et al.

G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package.
Eric Weisstein's World of Mathematics, Cycle Graph.
Eric Weisstein's World of Mathematics, Graph Loop.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A293311, A293312.

Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310 (magic labelings of LOOP X C_k, for k=1..9).

CoefficientList[Series[(1 + 112*z + 1983*z^2 + 9684*z^3 + 16120*z^4 + 9684*z^5 + 1983*z^6 + 112*z^7 + z^8)/(1 - z)^11, {z, 0, 22}], z]
LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {1, 123, 3281, 39175, 286555, 1508401, 6271378, 21836366, 66220705, 179784715, 445824731}, 25] (* Vincenzo Librandi, Oct 12 2017 *)

G.f.: (1 + 112*z + 1983*z^2 + 9684*z^3 + 16120*z^4 + 9684*z^5 + 1983*z^6 + 112*z^7 + z^8)/(1 - z)^11.

A293310 Number of magic labelings of the graph LOOP X C_9 (see comments) having magic sum n, n >= 0.

Original entry on oeis.org

1, 76, 1460, 13604, 81555, 363606, 1310974, 4029310, 10936124, 26868719, 60843972, 128724276, 257103166, 488789593, 890341484, 1562177132, 2651877099, 4371379686, 7018869628, 11006262508, 16893296453, 25429357976, 37604290362

Offset: 0

L. Edson Jeffery, Oct 06 2017

nonn

The graph LOOP X C_n is constructed by attaching a loop to each vertex of the cycle graph C_n.

The generating function for this sequence was found via the "Omega" package for Mathematica authored by Axel Riese. The package can be downloaded from the link given in the article by G. E. Andrews et al.

G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package.
Eric Weisstein's World of Mathematics, Cycle Graph.
Eric Weisstein's World of Mathematics, Graph Loop.

Cf. A293311, A293312.

Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293309 (magic labelings of LOOP X C_k, for k=1..8,10).

CoefficientList[Series[(1 + 67*z + 811*z^2 + 3049*z^3 + 4609*z^4 + 3049*z^5 + 811*z^6 + 67*z^7 + z^8)/((1 + z)*(1 - z)^10), {z, 0, 22}], z]

my(x='x+O('x^99));Vec((1+67*x+811*x^2+3049*x^3+4609*x^4+3049*x^5+811*x^6+67*x^7+x^8)/((1+x)*(1-x)^10)) \\ Altug Alkan, Oct 11 2017

G.f.: (1 + 67*z + 811*z^2 + 3049*z^3 + 4609*z^4 + 3049*z^5 + 811*z^6 + 67*z^7 + z^8)/((1 + z)*(1 - z)^10).

A064520 a(n) = + 1 - 2 - 3 + 4 + 5 + 6 - 7 - 8 - 9 - 10 + 11 + 12 + 13 + 14 + 15 - ... + (+-1)*n, where there is one plus, two minuses, three pluses, etc. (see A002024).

Original entry on oeis.org

1, -1, -4, 0, 5, 11, 4, -4, -13, -23, -12, 0, 13, 27, 42, 26, 9, -9, -28, -48, -69, -47, -24, 0, 25, 51, 78, 106, 77, 47, 16, -16, -49, -83, -118, -154, -117, -79, -40, 0, 41, 83, 126, 170, 215, 169, 122, 74, 25, -25, -76, -128, -181, -235, -290, -234, -177, -119, -60, 0, 61, 123, 186, 250, 315, 381, 314, 246, 177

Offset: 1

Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 07 2001

|a(n)| takes its locally maximal values when n is a triangular number, the maximal values being given by A019298.

The maximal positive/negative values occur for n = 1, 3, 6, 10, 15, 21 ... the triangular numbers and are a(n) = 1, -4, 11, -23, 42, -69,106, 215, 381, 616 ... +- int(sqrt(n^3/2) + 0.22098 * sqrt(n)). a(n) = n for n = 5, 13, 25, 41, 61, 85, ... m*(m*2-2)+1 and the previous number is equal to 0. Positive numbers which do not occur in this sequence are 2, 3, 6, 7, 8, 10, 12, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 43, 44, 45, 46, 48, ...

			a(9) = -13 because 1 - 2 - 3 + 4 + 5 + 6 - 7 - 8 - 9 = -13.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A002024, A019298, A064528.

a := proc(n) option remember: if n=1 then RETURN(1) fi: a(n-1) + n*(-1)^( floor(1/2 + sqrt(2*n)+1)); end: for n from 1 to 150 do printf(`%d,`,a(n)) od:

Accumulate[Flatten[Table[(-1)^(n+1) Range[(n(n-1))/2+1,(n(n+1))/2], {n,15}]]] (* Harvey P. Dale, Apr 22 2015 *)

t(n) = floor(1/2+sqrt(2*n))
for(n=1,200,print1(sum(k=1,n,(-1)^(t(k)+1)*k)," "))

t(n)= { floor(sqrt(2*n) + 1/2) }
{ for (n=1, 1000, a=sum(k=1, n, (-1)^(t(k) + 1)*k); write("b064520.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 17 2009

from math import isqrt
def A064520(n): return sum(k if (isqrt(k<<3)+1>>1)&1 else -k for k in range(1,n+1)) # Chai Wah Wu, Oct 16 2022

a(n) = Sum_{k=1..n} (-1)^(A002024(k)+1)*k.

More terms from James Sellers, Jason Earls and Vladeta Jovovic, Oct 08 2001

A236370 Sum of the largest parts in the partitions of 3n into 3 parts.

Original entry on oeis.org

1, 9, 34, 81, 163, 282, 454, 678, 973, 1335, 1786, 2319, 2959, 3696, 4558, 5532, 6649, 7893, 9298, 10845, 12571, 14454, 16534, 18786, 21253, 23907, 26794, 29883, 33223, 36780, 40606, 44664, 49009, 53601, 58498, 63657, 69139, 74898, 80998, 87390, 94141

Offset: 1

Wesley Ivan Hurt, Jan 23 2014

			Add first columns for a(n)..
                                               13 + 1 + 1
                                               12 + 2 + 1
                                               11 + 3 + 1
                                               10 + 4 + 1
                                                9 + 5 + 1
                                                8 + 6 + 1
                                                7 + 7 + 1
                                   10 + 1 + 1  11 + 2 + 2
                                    9 + 2 + 1  10 + 3 + 2
                                    8 + 3 + 1   9 + 4 + 2
                                    7 + 4 + 1   8 + 5 + 2
                                    6 + 5 + 1   7 + 6 + 2
                        7 + 1 + 1   8 + 2 + 2   9 + 3 + 3
                        6 + 2 + 1   7 + 3 + 2   8 + 4 + 3
                        5 + 3 + 1   6 + 4 + 2   7 + 5 + 3
                        4 + 4 + 1   5 + 5 + 2   6 + 6 + 3
            4 + 1 + 1   5 + 2 + 2   6 + 3 + 3   7 + 4 + 4
            3 + 2 + 1   4 + 3 + 2   5 + 4 + 3   6 + 5 + 4
1 + 1 + 1   2 + 2 + 2   3 + 3 + 3   4 + 4 + 4   5 + 5 + 5
   3(1)        3(2)        3(3)        3(4)        3(5)     ..   3n
---------------------------------------------------------------------
    1           9          34           81          163      ..  a(n)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A019298, A235988, A236364, A236758, A236762, A237264.

Table[3 n (n^2 - Floor[n^2/4]) - Sum[2 i^2 - Floor[i^2/4], {i, n}] -
  Sum[(n + i) (n - 2 i), {i, Floor[(n - 1)/2]}], {n, 100}]
LinearRecurrence[{2,1,-4,1,2,-1},{1,9,34,81,163,282},50] (* Harvey P. Dale, Nov 11 2017 *)

Vec(x*(2*x^4+8*x^3+15*x^2+7*x+1)/((x-1)^4*(x+1)^2) + O(x^100)) \\ Colin Barker, Jan 24 2014

a(n) = 3n * (n^2 - floor(n^2/4)) - Sum_{i=1..n} (2*i^2 - floor(i^2/4)) - Sum_{i=1..floor((n-1)/2)} (n + i) * (n - 2i).
From Colin Barker, Jan 24 2014: (Start)
a(n) = (-1+(-1)^n-(1+3*(-1)^n)*n-6*n^2+22*n^3)/16.
G.f.: x*(2*x^4+8*x^3+15*x^2+7*x+1) / ((x-1)^4*(x+1)^2). (End)
a(n) = Sum_{j=0..n-2} (Sum_{i=n+1+floor(j/2)-floor(1/j+1)..n+2*(j+1)} i), n > 1. - Wesley Ivan Hurt, Feb 10 2014
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6). - Wesley Ivan Hurt, Nov 19 2021

A237264 Number of partitions of 3n into 3 parts with largest part prime.

Original entry on oeis.org

0, 2, 4, 4, 8, 7, 13, 15, 22, 21, 28, 29, 36, 35, 44, 45, 54, 55, 67, 70, 83, 84, 96, 99, 116, 119, 135, 138, 154, 154, 170, 172, 187, 189, 208, 211, 231, 235, 259, 264, 285, 286, 306, 310, 334, 337, 361, 366, 389, 390, 413, 416, 441, 443, 468, 471, 496, 498

Offset: 1

Wesley Ivan Hurt, Feb 10 2014

			Count the primes in the first column for a(n).
                                               13 + 1 + 1
                                               12 + 2 + 1
                                               11 + 3 + 1
                                               10 + 4 + 1
                                                9 + 5 + 1
                                                8 + 6 + 1
                                                7 + 7 + 1
                                   10 + 1 + 1  11 + 2 + 2
                                    9 + 2 + 1  10 + 3 + 2
                                    8 + 3 + 1   9 + 4 + 2
                                    7 + 4 + 1   8 + 5 + 2
                                    6 + 5 + 1   7 + 6 + 2
                        7 + 1 + 1   8 + 2 + 2   9 + 3 + 3
                        6 + 2 + 1   7 + 3 + 2   8 + 4 + 3
                        5 + 3 + 1   6 + 4 + 2   7 + 5 + 3
                        4 + 4 + 1   5 + 5 + 2   6 + 6 + 3
            4 + 1 + 1   5 + 2 + 2   6 + 3 + 3   7 + 4 + 4
            3 + 2 + 1   4 + 3 + 2   5 + 4 + 3   6 + 5 + 4
1 + 1 + 1   2 + 2 + 2   3 + 3 + 3   4 + 4 + 4   5 + 5 + 5
   3(1)        3(2)        3(3)        3(4)        3(5)     ..   3n
---------------------------------------------------------------------
    0           2           4           4           8       ..  a(n)

Wesley Ivan Hurt, Table of n, a(n) for n = 1..58
Index entries for sequences related to partitions

Cf. A010051, A019298, A236364, A236370, A236758, A236762.

Table[Sum[Sum[PrimePi[i] - PrimePi[i - 1], {i, n + Floor[j/2] + 1 - Floor[1/(j + 1)], n + 2 (j + 1)}], {j, 0, n - 2}], {n, 50}]
Table[Count[IntegerPartitions[3 n,{3}],?(PrimeQ[#[[1]]]&)],{n,60}] (* _Harvey P. Dale, Mar 06 2022 *)

a(n) = Sum_{j=0..n-2} ( Sum_{i=n + 1 + floor(j/2) - floor(1/(j + 1))..n + 2(j + 1)} A010051(i) ).

A293312 Rectangular array read by antidiagonals: A(n,k) = tr((M_n)^k), k >= 0, where M_n is the n X n matrix M_1 = {{1}}, M_n = {{0,...,0,1},{0,...,0,1,1},...,{0,1,...,1},{1,...,1}}, n > 1, and tr(.) is the trace.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 3, 2, 4, 1, 4, 6, 2, 5, 1, 7, 11, 10, 3, 6, 1, 11, 26, 23, 15, 3, 7, 1, 18, 57, 70, 42, 21, 4, 8, 1, 29, 129, 197, 155, 69, 28, 4, 9, 1, 47, 289, 571, 533, 301, 106, 36, 5, 10, 1, 76, 650, 1640, 1884, 1223, 532, 154, 45, 5, 11

Offset: 1

L. Edson Jeffery, Oct 10 2017

Conjecture: For all n >= 1, for all k >= 2, A(n, k) = A293311(k, n); i.e., A(n, k) = number of magic labelings of the graph LOOP X C_k with magic sum n - 1.

			Array begins:
.   1 1  1   1    1     1      1       1       1        1         1
.   2 1  3   4    7    11     18      29      47       76       123
.   3 2  6  11   26    57    129     289     650     1460      3281
.   4 2 10  23   70   197    571    1640    4726    13604     39175
.   5 3 15  42  155   533   1884    6604   23219    81555    286555
.   6 3 21  69  301  1223   5103   21122   87677   363606   1508401
.   7 4 28 106  532  2494  11998   57271  274132  1310974   6271378
.   8 4 36 154  876  4654  25362  137155  743724  4029310  21836366
.   9 5 45 215 1365  8105  49347  298184 1806597 10936124  66220705
.  10 5 55 290 2035 13355  89848  599954 4016683 26868719 179784715
.  11 6 66 381 2926 21031 154935 1132942 8306078 60843972 445824731
.  ...

Cf. A293311.
Cf. A000012, A000032, A274975, A188128, A189237 (rows 1..5).
Cf. A000027, A000217, A019298, A006325, A244497, A244879, A244873, A244880, A293310, A293309 (columns k = 0,2..10 (conjectured)).

s[0, x_] := 1; s[1, x_] := x; s[k_, x_] := x*s[k - 1, x] - s[k - 2, x]; c[n_, j_] := 2 (-1)^(j - 1) Cos[j*Pi/(2 n + 1)]; a[n_, k_] := Round[Sum[s[n - 1, c[n, j]]^(k), {j, n}]];
(* Array: *)
Grid[Table[a[n, k], {n, 11}, {k, 0, 10}]]
(* Array antidiagonals flattened (gives this sequence): *)
Flatten[Table[a[n, k - n], {k, 11}, {n, k}]]

Let S(0, x) = 1, S(1, x) = x, S(k, x) = x*S(k - 1, x) - S(k - 2, x) (the S-polynomials of Wolfdieter Lang) and c(n, j) = 2*(-1)^(j - 1)*cos(j*Pi/(2*n + 1)). Then A(n, k) = Sum_{j=1..n} S(n - 1, c(n, j))^(k), n >= 1, k >= 0.

cp's OEIS Frontend

A236762 Number of partitions of 3n into 3 parts with the middle part prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A293311 Rectangular array read by antidiagonals: A(n,k) = number of magic labelings of the graph LOOP X C_n (see comments) having magic sum k, n >= 1, k >= 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A003438 Number of 5 X 5 matrices with nonnegative integer entries and row and column sums equal to n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A220075 Partial sums in rows of A220073, triangle read by rows.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

A293309 Number of magic labelings of the graph LOOP X C_10 (see comments) having magic sum n, n >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A293310 Number of magic labelings of the graph LOOP X C_9 (see comments) having magic sum n, n >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A064520 a(n) = + 1 - 2 - 3 + 4 + 5 + 6 - 7 - 8 - 9 - 10 + 11 + 12 + 13 + 14 + 15 - ... + (+-1)*n, where there is one plus, two minuses, three pluses, etc. (see A002024).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python