A201552 -id:A201552 - cp's OEIS frontend

A322549 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(n,k) is the constant term in the expansion of (Sum_{j=0..n} j*(x^j + x^(-j)))^k.

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 10, 0, 1, 0, 6, 12, 28, 0, 1, 0, 0, 198, 84, 60, 0, 1, 0, 20, 560, 2076, 324, 110, 0, 1, 0, 0, 5020, 14240, 12060, 924, 182, 0, 1, 0, 70, 20580, 213460, 146680, 49170, 2184, 280, 0, 1, 0, 0, 144774, 1984584, 3479700, 922680, 158418, 4536, 408, 0, 1

Offset: 0

Seiichi Manyama, Dec 15 2018

			Square array begins:
   1, 0,   0,    0,      0,       0,         0, ...
   1, 0,   2,    0,      6,       0,        20, ...
   1, 0,  10,   12,    198,     560,      5020, ...
   1, 0,  28,   84,   2076,   14240,    213460, ...
   1, 0,  60,  324,  12060,  146680,   3479700, ...
   1, 0, 110,  924,  49170,  922680,  32108060, ...
   1, 0, 182, 2184, 158418, 4226040, 203474180, ...

Seiichi Manyama, Antidiagonals n = 0..50, flattened

Columns 0-5: A000012, A000004, A006331, A303916, A305167, A318119.
Main diagonal gives A318793.
Cf. A201552.

A[0, 0] = 1; A[n_, k_] :=  Coefficient[Expand[Sum[j * (x^j + x^(-j)), {j, 0, n}]^k], x, 0]; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Amiram Eldar, Dec 16 2018 *)

T(n,k) = my(t=sum(j=0, n, j*(x^j + x^(-j)))^k); polcoef(numerator(t), poldegree(denominator(t))); \\ Michel Marcus, Dec 17 2018

A201553 Number of arrays of 6 integers in -n..n with sum zero.

Original entry on oeis.org

1, 141, 1751, 9331, 32661, 88913, 204763, 418503, 782153, 1363573, 2248575, 3543035, 5375005, 7896825, 11287235, 15753487, 21533457, 28897757, 38151847, 49638147, 63738149, 80874529, 101513259, 126165719, 155390809, 189797061, 230044751

Offset: 0

R. H. Hardin, Dec 02 2011

Row 6 of A201552.

			Some solutions for n=5:
..4....5....4...-2...-4....5...-1...-2...-1...-3...-3....0....2...-4....2...-5
..1...-4....5....3....4...-4....1....1....1....0....2...-2....1...-2...-1....1
.-2....3...-5....3....1....0...-4....2...-2....3....3....0....4....3....4....3
.-3...-3...-4....2....2...-3....5....4....4....0...-2....2....0....4...-1...-2
..5....4...-4...-2...-3...-1...-4...-1....1....0...-2....3...-4...-5...-2....4
.-5...-5....4...-4....0....3....3...-4...-3....0....2...-3...-3....4...-2...-1

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..210 from R. H. Hardin) [It was suggested that the initial terms of this b-file were wrong, but in fact they are correct. - _N. J. A. Sloane_, Jan 19 2019]
Robert Israel, Proof of "empirical" formula.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A201552.

a[n_] := Coefficient[Expand[Sum[x^k, {k, 0, 2n}]^6, x], x, 6n]; Array[a, 25, 0] (* Amiram Eldar, Dec 14 2018 *)

{a(n) = polcoeff((sum(k=0, 2*n, x^k))^6, 6*n, x)} \\ Seiichi Manyama, Dec 14 2018

Empirical: a(n) = (2*n+1)*(44*n^4+88*n^3+71*n^2+27*n+5)/5.
Empirical formula verified (see link) by Robert Israel, Dec 14 2018.
Empirical: a(n)= integral( (sin((n+1/2)x)/sin(x/2))^6, x=0..Pi)/Pi. - Yalcin Aktar, Dec 03 2011
Conjectures from Colin Barker, May 23 2018: (Start)
G.f.: x*(141 + 905*x + 940*x^2 + 120*x^3 + 7*x^4 - x^5) / (1 - x)^6.
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).
(End)
a(n) = [x^(6*n)] (Sum_{k=0..2*n} x^k)^6. - Seiichi Manyama, Dec 14 2018
E.g.f.: exp(x)*(5 + 700*x + 3675*x^2 + 3750*x^3 + 1100*x^4 + 88*x^5)/5. - Stefano Spezia, Sep 28 2024

a(0)=1 prepended by Seiichi Manyama, Dec 14 2018

A201554 Number of arrays of 7 integers in -n..n with sum zero.

Original entry on oeis.org

1, 393, 8135, 60691, 273127, 908755, 2473325, 5832765, 12354469, 24072133, 43874139, 75715487, 124853275, 198105727, 304134769, 453752153, 660249129, 939749665, 1311587215, 1798705035, 2428080047, 3231170251, 4244385685, 5509582933

Offset: 0

R. H. Hardin, Dec 02 2011

nonn

Row 7 of A201552.

			Some solutions for n=3:
..1....3....2....2....3...-2....0...-1...-2....1....0....1...-2...-3....1...-1
..3....2...-3....0...-2...-2....1...-3....1...-2....2....2....3....1....2...-1
.-3...-3....3....2...-2....1...-1....3...-3....3....1....1....0....0...-1....3
.-3...-2....2...-3....0....1....2....2...-1....1...-2...-3...-1....3....0....3
..0....0...-1....3...-1....1....2....1....1....1....1....2...-2...-1....0....2
.-1....0...-1...-1....2....3...-1...-1....1...-1...-2...-1....2....2...-2...-3
..3....0...-2...-3....0...-2...-3...-1....3...-3....0...-2....0...-2....0...-3

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..210 from R. H. Hardin) [It was suggested that the initial terms of this b-file were wrong, but in fact they are correct. - _N. J. A. Sloane_, Jan 19 2019]
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A201552.

a[n_] := Coefficient[Expand[Sum[x^k, {k, 0, 2n}]^7, x], x, 7n]; Array[a, 25, 0] (* Amiram Eldar, Dec 14 2018 *)

{a(n) = polcoeff((sum(k=0, 2*n, x^k))^7, 7*n, x)} \\ Seiichi Manyama, Dec 14 2018

Empirical: a(n) = 1+ 7*n*(n+1)*(841*n^4+1682*n^3+1568*n^2+727*n+222)/180.
Conjectures from Colin Barker, May 23 2018: (Start)
G.f.: (393 + 5384*x + 11999*x^2 + 5370*x^3 + 407*x^4 - 6*x^5 + x^6) / (1 - x)^7. -
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)
a(n) = [x^(7*n)] (Sum_{k=0..2*n} x^k)^7. - Seiichi Manyama, Dec 14 2018
Barker conjectures confirmed using technique similar to A201553.

a(0)=1 prepended by Seiichi Manyama, Dec 14 2018

A322535 Number of arrays of 8 integers in -n..n with sum zero.

Original entry on oeis.org

1, 1107, 38165, 398567, 2306025, 9377467, 30162301, 82073295, 197018321, 429042211, 864287973, 1633586615, 2927984825, 5017519755, 8273550157, 13194953119, 20438495649, 30853690355, 45522444469, 65803811463, 93384154505, 130333031003, 179165107485, 242908414063

Offset: 0

Seiichi Manyama, Dec 14 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Row 8 of A201552.

a[n_] := Coefficient[Expand[Sum[x^k, {k, 0, 2n}]^8, x], x, 8n]; Array[a, 25, 0] (* Amiram Eldar, Dec 14 2018 *)

{a(n) = polcoeff((sum(k=0, 2*n, x^k))^8, 8*n, x)}

a(n) = [x^(8*n)] (Sum_{k=0..2*n} x^k)^8.

a(n) = (2*n+1)*(9664*n^6+28992*n^5+37360*n^4+26400*n^3+10936*n^2+2568*n+315)/315.

A322536 Number of arrays of 9 integers in -n..n with sum zero.

Original entry on oeis.org

1, 3139, 180325, 2636263, 19610233, 97464799, 370487485, 1163205475, 3164588407, 7702189345, 17148949027, 35500063501, 69161990275, 128000343121, 226698100687, 386480229085, 637265493637, 1020310909975, 1591418959705, 2424782370859, 3617545938373, 5295169534843

Offset: 0

Seiichi Manyama, Dec 14 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9, -36, 84, -126, 126, -84, 36, -9, 1).

Row 9 of A201552.

a[n_] := Coefficient[Expand[Sum[x^k, {k, 0, 2n}]^9, x], x, 9n]; Array[a, 25, 0] (* Amiram Eldar, Dec 14 2018 *)

{a(n) = polcoeff((sum(k=0, 2*n, x^k))^9, 9*n, x)}

a(n) = [x^(9*n)] (Sum_{k=0..2*n} x^k)^9.

A322537 Number of arrays of 10 integers in -n..n with sum zero.

Original entry on oeis.org

1, 8953, 856945, 17538157, 167729959, 1018872811, 4577127763, 16581420835, 51125645317, 139071924069, 342237634221, 775938666273, 1643151128475, 3284313415527, 6247630238079, 11385659856231, 19984965376233, 33936690554865, 55957080110537, 89868204063989, 140950465124591

Offset: 0

Seiichi Manyama, Dec 14 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (10, -45, 120, -210, 252, -210, 120, -45, 10, -1).

Row 10 of A201552.

a[n_] := Coefficient[Expand[Sum[x^k, {k, 0, 2n}]^10, x], x, 10n]; Array[a, 25, 0] (* Amiram Eldar, Dec 14 2018 *)

{a(n) = polcoeff((sum(k=0, 2*n, x^k))^10, 10*n, x)}

a(n) = [x^(10*n)] (Sum_{k=0..2*n} x^k)^10.

A322538 Number of arrays of n integers in -8..8 with sum zero.

Original entry on oeis.org

1, 1, 17, 217, 3281, 50101, 782153, 12354469, 197018321, 3164588407, 51125645317, 829858058019, 13522838817929, 221088393889403, 3624855419439957, 59576725303913167, 981272544393935569, 16192709833199300337, 267654541150392845543, 4430755975190532983531

Offset: 0

Seiichi Manyama, Dec 14 2018

Seiichi Manyama, Table of n, a(n) for n = 0..350

Column 8 of A201552.

seq(add((-1)^k*binomial(n, k)*binomial(9*n-17*k-1, n-1), k = 0..floor(n/2)), n = 0..20); # Peter Bala, Oct 19 2024

a[n_] := If[n==0, 1, Coefficient[Expand[Sum[x^k, {k, 0, 16}]^n], x^(8n)]]; Array[a, 25, 0] (* Amiram Eldar, Dec 14 2018 *)

{a(n) = polcoeff((sum(k=0, 16, x^k))^n, 8*n, x)}

a(n) = [x^(8*n)] (Sum_{k=0..16} x^k)^n.

a(n) ~ 17^n / (4*sqrt(3*Pi*n)). - Vaclav Kotesovec, Dec 15 2018

A322539 Number of arrays of n integers in -9..9 with sum zero.

Original entry on oeis.org

1, 1, 19, 271, 4579, 78151, 1363573, 24072133, 429042211, 7702189345, 139071924069, 2522948398895, 45949039890469, 839611990929219, 15385356833972711, 282616668487409521, 5202536118941844771, 95950964483217949751, 1772592132899627652691, 32795665902734099555845

Offset: 0

Seiichi Manyama, Dec 14 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..300

Column 9 of A201552.

seq(add((-1)^k*binomial(n, k)*binomial(10*n-19*k-1, n-1), k = 0..floor(n/2)), n = 0..20); # Peter Bala, Oct 19 2024

a[n_] := If[n==0, 1, Coefficient[Expand[Sum[x^k, {k, 0, 18}]^n], x^(9n)]]; Array[a, 25, 0] (* Amiram Eldar, Dec 14 2018 *)

{a(n) = polcoeff((sum(k=0, 18, x^k))^n, 9*n, x)}

a(n) = [x^(9*n)] (Sum_{k=0..18} x^k)^n.

a(n) ~ 19^n / (2*sqrt(15*Pi*n)). - Vaclav Kotesovec, Dec 15 2018

A322540 Number of arrays of n integers in -10..10 with sum zero.

Original entry on oeis.org

1, 1, 21, 331, 6181, 116601, 2248575, 43874139, 864287973, 17148949027, 342237634221, 6862175029689, 138132246844879, 2789732439723061, 56501091296763855, 1147129435985083731, 23339695295337149925, 475767618472211229549, 9714486212587875804351

Offset: 0

Seiichi Manyama, Dec 14 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..300

Column 10 of A201552.

seq(add((-1)^k*binomial(n, k)*binomial(11*n-21*k-1, n-1), k = 0..floor(n/2)), n = 0..20); # Peter Bala, Oct 19 2024

a[n_] := If[n==0, 1, Coefficient[Expand[Sum[x^k, {k, 0, 20}]^n], x^(10n)]]; Array[a, 25, 0] (* Amiram Eldar, Dec 14 2018 *)

{a(n) = polcoeff((sum(k=0, 20, x^k))^n, 10*n, x)}

a(n) = [x^(10*n)] (Sum_{k=0..20} x^k)^n.

a(n) ~ sqrt(3) * 21^n / (2*sqrt(55*Pi*n)). - Vaclav Kotesovec, Dec 15 2018

cp's OEIS Frontend

A322549 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(n,k) is the constant term in the expansion of (Sum_{j=0..n} j*(x^j + x^(-j)))^k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A201553 Number of arrays of 6 integers in -n..n with sum zero.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A201554 Number of arrays of 7 integers in -n..n with sum zero.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A322535 Number of arrays of 8 integers in -n..n with sum zero.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A322536 Number of arrays of 9 integers in -n..n with sum zero.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A322537 Number of arrays of 10 integers in -n..n with sum zero.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A322538 Number of arrays of n integers in -8..8 with sum zero.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A322539 Number of arrays of n integers in -9..9 with sum zero.

Views