cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A066535 Number of ways of writing n as a sum of n squares.

Original entry on oeis.org

1, 2, 4, 8, 24, 112, 544, 2368, 9328, 34802, 129064, 491768, 1938336, 7801744, 31553344, 127083328, 509145568, 2035437440, 8148505828, 32728127192, 131880275664, 532597541344, 2153312518240, 8710505815360, 35250721087168, 142743029326162, 578472382307304
Offset: 0

Views

Author

Peter Bertok (peter(AT)bertok.com), Jan 07 2002

Keywords

Examples

			There are a(3) = 8 solutions (x,y,z) of 3 = x^2 + y^2 + z^2: (1,1,1), (-1,-1,-1), 3 permutations of (1,1,-1) and 3 permutations of (1,-1,-1).

Links

Crossrefs

Cf. A004018, A005875, A000118, A066536.
Cf. A122141, A166952. - Paul D. Hanna, Oct 25 2009
a(n^2) gives A361431.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<0 or t<1, 0,
      b(n, t-1) +2*add(b(n-j^2, t-1), j=1..isqrt(n))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 16 2014
  • Mathematica
    Join[{1}, Table[SquaresR[n, n], {n, 24}]]
  • PARI
    {a(n)=local(THETA3=1+2*sum(k=1,sqrtint(n),x^(k^2))+x*O(x^n)); polcoeff(THETA3^n, n)} /* Paul D. Hanna, Oct 25 2009 */

Formula

a(n) equals the coefficient of x^n in the n-th power of Jacobi theta_3(x) where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). - Paul D. Hanna, Oct 25 2009
a(n) ~ c * d^n / sqrt(n), where d = 4.13273137623493996302796465... (= 1/radius of convergence A166952), c = 0.2820942036723951157919967... . - Vaclav Kotesovec, Sep 12 2014

Extensions

Edited by Dean Hickerson, Jan 12 2002
a(0) added by Paul D. Hanna, Oct 25 2009
Edited by R. J. Mathar, Oct 29 2009

A379200 G.f. A(x,y) satisfies 1/x = Sum_{n=-oo..+oo} A(x,y)^n * (A(x,y)^n + y)^(n+1), as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y), read by rows.

Original entry on oeis.org

1, 2, 1, 4, 4, 2, 8, 13, 12, 5, 18, 40, 52, 40, 14, 52, 130, 204, 215, 140, 42, 184, 472, 813, 1004, 896, 504, 132, 688, 1863, 3430, 4588, 4816, 3738, 1848, 429, 2512, 7536, 15016, 21472, 24540, 22656, 15576, 6864, 1430, 8866, 30144, 65880, 102177, 124830, 126801, 104940, 64779, 25740, 4862, 30824, 118420, 284305, 483300, 636750, 693528, 638825, 479908, 268840, 97240, 16796
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2024

Keywords

Comments

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.

Examples

			G.f.: A(x,y) = x*(1) + x^2*(2 + y) + x^3*(4 + 4*y + 2*y^2) + x^4*(8 + 13*y + 12*y^2 + 5*y^3) + x^5*(18 + 40*y + 52*y^2 + 40*y^3 + 14*y^4) + x^6*(52 + 130*y + 204*y^2 + 215*y^3 + 140*y^4 + 42*y^5) + x^7*(184 + 472*y + 813*y^2 + 1004*y^3 + 896*y^4 + 504*y^5 + 132*y^6) + x^8*(688 + 1863*y + 3430*y^2 + 4588*y^3 + 4816*y^4 + 3738*y^5 + 1848*y^6 + 429*y^7) + x^9*(2512 + 7536*y + 15016*y^2 + 21472*y^3 + 24540*y^4 + 22656*y^5 + 15576*y^6 + 6864*y^7 + 1430*y^8) + x^10*(8866 + 30144*y + 65880*y^2 + 102177*y^3 + 124830*y^4 + 126801*y^5 + 104940*y^6 + 64779*y^7 + 25740*y^8 + 4862*y^9) + ...
where 1/x = Sum_{n=-oo..+oo} A(x,y)^n * (A(x,y)^n + y)^(n+1).
TRIANGLE.
This triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for n >= 1, k=0..n-1, begins
n = 1: [1];
n = 2: [2, 1];
n = 3: [4, 4, 2];
n = 4: [8, 13, 12, 5];
n = 5: [18, 40, 52, 40, 14];
n = 6: [52, 130, 204, 215, 140, 42];
n = 7: [184, 472, 813, 1004, 896, 504, 132];
n = 8: [688, 1863, 3430, 4588, 4816, 3738, 1848, 429];
n = 9: [2512, 7536, 15016, 21472, 24540, 22656, 15576, 6864, 1430];
n =10: [8866, 30144, 65880, 102177, 124830, 126801, 104940, 64779, 25740, 4862];
n =11: [30824, 118420, 284305, 483300, 636750, 693528, 638825, 479908, 268840, 97240, 16796];
n =12: [108088, 460746, 1205402, 2242581, 3213584, 3758727, 3731794, 3154866, 2171312, 1113398, 369512, 58786];
  ...
RELATED SEQUENCES.
A000108(n) = T(n+1,n) for n >= 0 (Catalan numbers).
A028329(n) = T(n+2,n) for n >= 0.
A166952(n) = T(n+1,0) for n >= 0 (g.f. F(x) = theta_3(x*F(x))).
A379201(n) = T(n,1) for n >= 2 (column 1).
A379206(n) = T(2*n-1,n-1) for n >= 1 (central terms).
A378264(n) = Sum_{k=0..n-1} T(n,k) for n >= 1.
A379199(n) = Sum_{k=0..n-1} T(n,k) * (-1)^k for n >= 1.
A379202(n) = Sum_{k=0..n-1} T(n,k) * 2^k for n >= 1.
A379203(n) = Sum_{k=0..n-1} T(n,k) * 3^k for n >= 1.
A379204(n) = Sum_{k=0..n-1} T(n,k) * 4^k for n >= 1.
A379205(n) = Sum_{k=0..n-1} T(n,k) * 5^k for n >= 1.
ALTERNATIVE FORMAT.
This triangle may also be presented as a rectangular table like so:
[  1,    1,     2,      5,     14,      42,      132, ...];
[  2,    4,    12,     40,    140,     504,     1848, ...];
[  4,   13,    52,    215,    896,    3738,    15576, ...];
[  8,   40,   204,   1004,   4816,   22656,   104940, ...];
[ 18,  130,   813,   4588,  24540,  126801,   638825, ...];
[ 52,  472,  3430,  21472, 124830,  693528,  3731794, ...];
[184, 1863, 15016, 102177, 636750, 3758727, 21365548, ...];
...

Links

Crossrefs

Cf. A166952 (column 0, y=0), A378264 (row sums), A379201 (column 1), A379206 (central terms).
Cf. A379199 (y=-1), A379202 (y=2), A379203 (y=3), A379204 (y=4), A379205 (y=5).
Cf. A000108 (main diagonal), A028329 (diagonal).

Programs

  • PARI
    {T(n,k) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + y)^(m+1) ), #V-3); ); polcoef(polcoef(A, n, x), k, y)}
for(n=1,12, for(k=0,n-1, print1(T(n,k),", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=1} Sum_{k=0..n-1} T(n,k)*x^n*y^k satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x,y)^n * (A(x,y)^n + y)^(n+1).
(2) 1/x = Sum_{n=-oo..+oo} A(x,y)^(2*n) * (A(x,y)^n - y)^n.
(3) A(x,y) = x * Sum_{n=-oo..+oo} A(x,y)^(n^2) / (1 + y*A(x,y)^(n+1))^n.
(4) A(x,y) = x * Sum_{n=-oo..+oo} A(x,y)^(n^2) / (1 - y*A(x,y)^(n+1))^(n+1).
(5) A(B(x,y), y) = x where B(x,y) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + y)^(n+1) ).

A378264 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (1 + A(x)^n)^(n+1).

Original entry on oeis.org

1, 3, 10, 38, 164, 783, 4005, 21400, 117602, 659019, 3748736, 21588796, 125646501, 737977155, 4369147468, 26048215099, 156249597852, 942344615209, 5710710976884, 34756875588376, 212361179832431, 1302068876523950, 8009024360554817, 49407447276951470, 305609996146288873, 1895015255546957578
Offset: 1

Views

Author

Paul D. Hanna, Dec 08 2024

Keywords

Comments

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.

Examples

			G.f.: A(x) = x + 3*x^2 + 10*x^3 + 38*x^4 + 164*x^5 + 783*x^6 + 4005*x^7 + 21400*x^8 + 117602*x^9 + 659019*x^10 + 3748736*x^11 + 21588796*x^12 + ...
SPECIFIC VALUES.
A(t) = 1/3 at t = 0.14832728317680424382350400745104642263167027946862...
A(t) = 1/4 at t = 0.13433913917600443178696714330960568436967435856815...
A(t) = 1/5 at t = 0.12029812285398972879219940261295281978412524937754...
A(3/20) = 0.3521325903099608361455770617898033111722103407971...
A(1/7) = 0.29252723487814042698570516039406838227427731852655...
A(1/8) = 0.21500724214149512130643660913381998900575603076452...
A(1/9) = 0.17407688053908806913569913139334508111874650183559...
A(1/10) = 0.14711097488062849474543678333471254427936118296317...

Links

Crossrefs

Cf. A379200, A379199, A166952, A379202, A379203, A379204, A379205.

Programs

  • PARI
    {a(n) = my(V=[0,1],A); for(i=1,n, V=concat(V,0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A,#A, A^m*(1 + A^m)^(m+1) ), #V-3); ); polcoef(A,n)}
for(n=1,40,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (1 + A(x)^n)^(n+1).
(2) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + A(x)^(n+1))^n.
From Paul D. Hanna, Dec 20 2024: (Start)
(3) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 1)^n.
(4) A(x) = x * Sum_{n=-oo..+oo, n <> -1} A(x)^(n^2) / (1 - A(x)^(n+1))^(n+1).
(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 1)^(n+1) ).
(End)

A379199 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n - 1)^(n+1).

Original entry on oeis.org

1, 1, 2, 2, 4, 9, 45, 164, 546, 1493, 3944, 10588, 32997, 112945, 396404, 1330461, 4265180, 13292275, 41778612, 135378928, 452828655, 1534394542, 5175561385, 17246318586, 56998526633, 188492707958, 628391304843, 2115131897264, 7162685531894, 24280930956521, 82152859633099
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2024

Keywords

Comments

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 45*x^7 + 164*x^8 + 546*x^9 + 1493*x^10 + 3944*x^11 + 10588*x^12 + ...
SPECIFIC VALUES.
A(t) = 1/2 at t = 0.28045847462385815185359630099816126187110099265378...
  where t = 1/Sum_{n=-oo..+oo} (-1)^n * (2^(n-1) - 1)^n / 2^(n^2-1),
  also, t = 1/Sum_{n=-oo..+oo} (2^(n-1) + 1)^(n-1) / 2^(n^2-1).
A(t) = 1/3 at t = 0.23482705460970305955617199360925350115096428519729...
  where t = 1/Sum_{n=-oo..+oo} (-1)^n * (3^(n-1) - 1)^n / 3^(n^2-1),
  also, t = 1/Sum_{n=-oo..+oo} (3^(n-1) + 1)^(n-1) / 3^(n^2-1).
A(t) = 1/4 at t = 0.19291797602834900465339136778069433360676297133766...
  where t = 1/Sum_{n=-oo..+oo} (-1)^n * (4^(n-1) - 1)^n / 4^(n^2-1),
  also, t = 1/Sum_{n=-oo..+oo} (4^(n-1) + 1)^(n-1) / 4^(n^2-1).
A(1/4) = 0.37094847513809700088242935848658292140487254454012...
  where 4 = Sum_{n=-oo..+oo} A(1/4)^n * (A(1/4)^n - 1)^(n+1),
  also, 4 = Sum_{n=-oo..+oo} A(1/4)^(2*n) * (A(1/4)^n + 1)^n.
A(1/5) = 0.26269124124750053890427847522296583687631694884657...
A(1/6) = 0.20631303406093749454201994379654348907240460444958...
A(1/7) = 0.17034902087146833005156413354158308643804109633470...
A(1/8) = 0.14521334319041207588863463072178319621820854479438...

Links

Crossrefs

Cf. A379200, A166952, A378264, A379202, A379203, A379204, A379205.

Programs

  • PARI
    {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m - 1)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n - 1)^(n+1).
(2) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n + 1)^n.
(3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + A(x)^(n+1))^(n+1).
(4) A(x) = x * Sum_{n=-oo..+oo, n <> -1} A(x)^(n^2) / (1 - A(x)^(n+1))^n.
(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n - 1)^(n+1) ).

A379202 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 2)^(n+1).

Original entry on oeis.org

1, 4, 20, 122, 850, 6432, 51324, 424694, 3608592, 31291658, 275774228, 2462835772, 22239367632, 202713590686, 1862689951724, 17235880764264, 160466865121154, 1502055108051124, 14127846520455180, 133455751612975948, 1265563747442829216, 12043611154775588194, 114978748131733714360
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2024

Keywords

Comments

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.
Conjecture: a(n) is even for n > 1.
It appears that a(n) == 2 (mod 4) at n = A028309(k) for k >= 4.

Examples

			G.f.: A(x) = x + 4*x^2 + 20*x^3 + 122*x^4 + 850*x^5 + 6432*x^6 + 51324*x^7 + 424694*x^8 + 3608592*x^9 + 31291658*x^10 + ...
SPECIFIC VALUES.
A(t) = 1/6 at t = 0.090270773138940793847220645261976952310511883470512...
  where t = 1/Sum_{n=-oo..+oo} (1 + 2*6^(n-1))^n / 6^(n^2-1).
A(t) = 1/7 at t = 0.084362907984862824662513569761745773472320783010611...
  where t = 1/Sum_{n=-oo..+oo} (1 + 2*7^(n-1))^n / 7^(n^2-1).
A(t) = 1/8 at t = 0.078703999402417120618295617221021413542415048822164...
  where t = 1/Sum_{n=-oo..+oo} (1 + 2*8^(n-1))^n / 8^(n^2-1).
A(1/11) = 0.16976727159020613475135380983780463368461713164010...
A(1/12) = 0.13933682309394427848416123650354034389806333559384...
A(1/15) = 0.09515898887066227963795425335824195002284059150209...
A(1/20) = 0.06369786461564277053938913595571090186089127528505...

Links

Crossrefs

Cf. A379200, A379199, A166952, A378264, A379203, A379204, A379205.

Programs

  • PARI
    {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 2)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 2)^(n+1).
(2) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 2)^n.
(3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 2*A(x)^(n+1))^n.
(4) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 2*A(x)^(n+1))^(n+1).
(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 2)^(n+1) ).

A379203 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 3)^(n+1).

Original entry on oeis.org

1, 5, 34, 290, 2820, 29629, 327301, 3744868, 43981858, 527126689, 6420981368, 79260797860, 989306411413, 12464737320229, 158320378037652, 2025016002188169, 26060398562711196, 337197048402240367, 4384067953773647268, 57245716462267462224, 750403639664344374239, 9871281245683966836462
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2024

Keywords

Comments

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.

Examples

			G.f.: A(x) = x + 5*x^2 + 34*x^3 + 290*x^4 + 2820*x^5 + 29629*x^6 + 327301*x^7 + 3744868*x^8 + 43981858*x^9 + 527126689*x^10 + ...
SPECIFIC VALUES.
A(t) = 1/7 at t = 0.069769772400266469707360138034033927488705716660080...
  where t = 1/Sum_{n=-oo..+oo} (1 + 3*7^(n-1))^n / 7^(n^2-1).
A(t) = 1/8 at t = 0.067295105779482404156544832668824160420208234924667...
  where t = 1/Sum_{n=-oo..+oo} (1 + 3*8^(n-1))^n / 8^(n^2-1).
A(t) = 1/9 at t = 0.064327556053208007320009998534415581932268509899202...
  where t = 1/Sum_{n=-oo..+oo} (1 + 3*9^(n-1))^n / 9^(n^2-1).
A(t) = 1/10 at t = 0.06126924119589872239866986020862532219839002819792...
  where t = 1/Sum_{n=-oo..+oo} (1 + 3*10^(n-1))^n / 10^(n^2-1).
A(1/15) = 0.12166176397390884847529063617720403039492284665035...
A(1/16) = 0.10420546336336096378642246758350885785023968035181...
A(1/20) = 0.07053009254165709187694647754531300907207762301254...

Links

Crossrefs

Cf. A379200, A379199, A166952, A378264, A379202, A379204, A379205.

Programs

  • PARI
    {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 3)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 3)^(n+1).
(2) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 3)^n.
(3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 3*A(x)^(n+1))^n.
(4) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 3*A(x)^(n+1))^(n+1).
(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 3)^(n+1) ).

A379204 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 4)^(n+1).

Original entry on oeis.org

1, 6, 52, 572, 7154, 96444, 1365480, 20015404, 301104656, 4622137698, 72110068424, 1140008607808, 18223311950352, 294049155429240, 4783093039542544, 78348659072215696, 1291254702576739650, 21396346604365855060, 356250789435149406252, 5957201829333106382128, 100003077199160840926640
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2024

Keywords

Comments

Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.
Conjecture: a(n) is even for n > 1.
Conjecture: a(n) == 2 (mod 4) iff n = (k-1)^2 + 1 for some k > 1.

Examples

			G.f.: A(x) = x + 6*x^2 + 52*x^3 + 572*x^4 + 7154*x^5 + 96444*x^6 + 1365480*x^7 + 20015404*x^8 + 301104656*x^9 + 4622137698*x^10 + ...

Links

Crossrefs

Cf. A379200, A379199, A166952, A378264, A379202, A379203, A379205.

Programs

  • PARI
    {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 4)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 4)^(n+1).
(2) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 4)^n.
(3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 4*A(x)^(n+1))^n.
(4) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 4*A(x)^(n+1))^(n+1).
(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 4)^(n+1) ).

A379205 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 5)^(n+1).

Original entry on oeis.org

1, 7, 74, 998, 15268, 251427, 4345869, 77751128, 1427455842, 26740178711, 509068777424, 9820550568868, 191554931918517, 3771529984556599, 74857068226445132, 1496158969938529383, 30086862802675119068, 608303992207446069349, 12358069554479794052292, 252144178158939689795128
Offset: 1

Views

Author

Paul D. Hanna, Dec 20 2024

Keywords

Examples

			G.f.: A(x) = x + 7*x^2 + 74*x^3 + 998*x^4 + 15268*x^5 + 251427*x^6 + 4345869*x^7 + 77751128*x^8 + 1427455842*x^9 + 26740178711*x^10 + ...

Links

Crossrefs

Cf. A379200, A379199, A166952, A378264, A379202, A379203, A379204.

Programs

  • PARI
    {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(A^m + 5)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 5)^(n+1).
(2) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 5)^n.
(3) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + 5*A(x)^(n+1))^n.
(4) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 - 5*A(x)^(n+1))^(n+1).
(5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 5)^(n+1) ).

A302860 a(n) = [x^n] theta_3(x)^n/(1 - x), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 3, 9, 27, 89, 333, 1341, 5449, 21697, 84663, 327829, 1275739, 5020457, 19964623, 79883141, 320317827, 1284656385, 5152761033, 20686311261, 83182322509, 335110196569, 1352277390001, 5463873556381, 22097867887045, 89441286136465, 362277846495883, 1468465431530457
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 14 2018

Keywords

Comments

a(n) = number of integer lattice points inside the n-dimensional hypersphere of radius sqrt(n).

Links

Crossrefs

Main diagonal of A122510.
Cf. A000122, A001650, A046895, A057655, A066535, A117609, A302861, A066536, A166952.

Programs

  • Mathematica
    Table[SeriesCoefficient[EllipticTheta[3, 0, x]^n/(1 - x), {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x) Sum[x^k^2, {k, -n, n}]^n, {x, 0, n}], {n, 0, 26}]

Formula

a(n) = A122510(n,n).
a(n) ~ c / (sqrt(n) * r^n), where r = 0.241970723224463308846762732757915397312... (= radius of convergence A166952) and c = 0.716940866073606328... - Vaclav Kotesovec, Apr 14 2018

A066536 Number of ways of writing n as a sum of n+1 squares.

Original entry on oeis.org

1, 4, 12, 32, 90, 312, 1288, 5504, 22608, 88660, 339064, 1297056, 5043376, 19975256, 80027280, 321692928, 1291650786, 5177295432, 20748447108, 83279292960, 335056780464, 1351064867328, 5456890474248, 22063059606912
Offset: 0

Views

Author

Peter Bertok (peter(AT)bertok.com), Jan 07 2002

Keywords

Examples

			There are a(2)=12 solutions (x,y,z) of 2=x^2+y^2+z^2: 3 permutations of (1,1,0), 3 permutations of (-1,-1,0) and 6 permutations of (1, -1,0).

Links

Crossrefs

Cf. A004018, A005875, A000118, A066535.
Cf. A122141, A166952. - Paul D. Hanna, Oct 26 2009

Programs

  • Mathematica
    Join[{1}, Table[SquaresR[n+1, n], {n, 24}]]
(* Calculation of constants {d,c}: *) {1/r, Sqrt[s/(2*Pi*r^2*Derivative[0, 0, 2][EllipticTheta][3, 0, r*s])]} /. FindRoot[{s == EllipticTheta[3, 0, r*s], r*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s] == 1}, {r, 1/4}, {s, 5/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Nov 16 2023 *)
  • PARI
    {a(n)=local(THETA3=1+2*sum(k=1,sqrtint(n),x^(k^2))+x*O(x^n));polcoeff(THETA3^(n+1), n)} /* Paul D. Hanna, Oct 26 2009*/

Formula

a(n) equals the coefficient of x^n in the (n+1)-th power of Jacobi theta_3(x) where theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2). - Paul D. Hanna, Oct 26 2009
a(n) is divisible by n+1: a(n)/(n+1) = A166952(n) for n>=0. - Paul D. Hanna, Oct 26 2009
a(n) ~ c * d^n / sqrt(n), where d = 4.13273137623493996302796465... (= 1/radius of convergence A166952), c = 0.70710538549959357505200... . - Vaclav Kotesovec, Sep 10 2014

Extensions

Edited by Dean Hickerson, Jan 12 2002
a(0) added by Paul D. Hanna, Oct 26 2009
Edited by R. J. Mathar, Oct 29 2009
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