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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A381277 Expansion of e.g.f. exp(sinh(3*x) / 3).

Original entry on oeis.org

1, 1, 1, 10, 37, 172, 1477, 8416, 74377, 683344, 5836969, 67102048, 699721453, 8268521536, 107106298093, 1347611617792, 19462095444241, 279380302430464, 4247519795325649, 68946703997616640, 1122787065355425973, 19697500164381137920, 351304020205694058133
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Crossrefs

Cf. A003724, A009229.
Cf. A136630.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, 3^(n-k)*a136630(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 9^k * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} 3^(n-k) * A136630(n,k).

A381285 Expansion of e.g.f. 1/(1 - sin(2*x) / 2).

Original entry on oeis.org

1, 1, 2, 2, -8, -104, -688, -3088, -128, 209536, 3145472, 29795072, 139389952, -1715047424, -60056147968, -1004215072768, -10305404960768, -1945682345984, 2949643589844992, 84438462955323392, 1458284922371571712, 12032890515685113856, -245515800089314459648
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Crossrefs

Cf. A000111, A381286.
Cf. A136630.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1-Sin[2x]/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 09 2025 *)
  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*(2*I)^(n-k)*a136630(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-4)^k * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * (2*i)^(n-k) * A136630(n,k), where i is the imaginary unit.

A381382 E.g.f. A(x) satisfies A(x) = 1/( 1 - sinh(x * A(x)^2) / A(x)^2 ).

Original entry on oeis.org

1, 1, 2, 7, 48, 541, 7600, 120891, 2178176, 45053401, 1065957888, 28344376303, 831973593088, 26647344263541, 925300511922176, 34668496386129763, 1394928344160731136, 59986286728056665905, 2744940504174063714304, 133158543838350039763671
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2025

Keywords

Crossrefs

Cf. A364980, A381376, A381378, A381384.
Cf. A136630.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n-k+1, k)/(2*n-k+1)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(2*n-k+1,k)/(2*n-k+1) * A136630(n,k).

A381384 E.g.f. A(x) satisfies A(x) = 1/( 1 - sin(x * A(x)^2) / A(x)^2 ).

Original entry on oeis.org

1, 1, 2, 5, 0, -299, -5840, -90791, -1210496, -11174519, 71397888, 8367496301, 327020705792, 9709296136541, 226223975684096, 2946493117173761, -87437164233621504, -9675847870039338095, -535455805780063748096, -22518479178045130002731, -706013052362778282033152
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2025

Keywords

Crossrefs

Cf. A364980, A381376, A381378, A381382.
Cf. A136630.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n-k+1, k)/(2*n-k+1)*I^(n-k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(2*n-k+1,k)/(2*n-k+1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A248836 Number of length n arrays x(i), i=1..n with x(i) in 0..i and no value appearing more than 2 times.

Original entry on oeis.org

1, 2, 6, 22, 96, 482, 2736, 17302, 120576, 917762, 7574016, 67354582, 642041856, 6530291042, 70589700096, 808090395862, 9766250151936, 124258689304322, 1660195646078976, 23239748527125142, 340125128186658816, 5194627679316741602, 82645634692238278656
Offset: 0

Views

Author

R. H. Hardin, Oct 15 2014

Keywords

Comments

Column 2 of A248842

Examples

			Some solutions for n=6
..1....1....1....1....1....0....1....0....0....0....1....0....1....1....1....0
..0....2....2....0....2....2....2....2....2....1....1....1....2....0....0....1
..0....3....0....1....2....1....2....1....1....1....0....3....0....1....2....0
..1....4....0....0....4....2....0....3....3....2....3....3....4....0....0....2
..4....0....3....4....3....4....4....0....1....5....4....2....2....2....2....4
..5....6....4....5....4....0....3....6....6....5....0....2....5....4....4....4

Links

Crossrefs

Cf. A000111, A248842.

Formula

From Seiichi Manyama, Feb 17 2025: (Start)
Conjecture: E.g.f.: 1/(1 - sin(x))^2.
If the above conjecture is correct, the following general term is obtained:
a(n) = Sum_{k=0..n} (k+1)! * i^(n-k) * A136630(n,k), where i is the imaginary unit. (End)
Conjecture from Mikhail Kurkov, Jun 26 2025: (Start)
a(n) = R(n+1,0) where
R(0,0) = 1,
R(n,k) = Sum_{j=0..n-k-1} R(n-1,j) for 0 <= k < n,
R(n,n) = Sum_{j=0..n-1} R(n,j). (End)

Extensions

a(0)=1 prepended by Seiichi Manyama, Feb 17 2025

A351891 Expansion of e.g.f. exp( sinh(sqrt(2)*x) / sqrt(2) ).

Original entry on oeis.org

1, 1, 1, 3, 9, 25, 105, 443, 1969, 10609, 57265, 338547, 2190969, 14498185, 104277849, 784965803, 6150938593, 51229928929, 440694547681, 3967606065891, 37247506348905, 361022009762809, 3645855348771273, 38001754007842715, 409302848055407761, 4558828622414199121
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 24 2022

Keywords

Crossrefs

Cf. A003724, A009229, A055882, A136630, A351892, A381277.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Sinh[Sqrt[2] x]/Sqrt[2]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 2 k] 2^k a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}]

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * 2^k * a(n-2*k-1).
a(n) = Sum_{k=0..n} 2^((n-k)/2) * A136630(n,k). - Seiichi Manyama, Feb 20 2025

A351892 Expansion of e.g.f. exp( sinh(sqrt(3)*x) / sqrt(3) ).

Original entry on oeis.org

1, 1, 1, 4, 13, 40, 205, 952, 4921, 31168, 189145, 1318528, 9843781, 74869888, 632536933, 5475991552, 49996774897, 485393809408, 4829958877105, 50858117779456, 554544498995965, 6259096187060224, 73822470722135293, 894846287081242624, 11261265009125680681, 146272258394568687616
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 24 2022

Keywords

Crossrefs

Cf. A003724, A009229, A136630, A247452, A351891, A381277.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Sinh[Sqrt[3] x]/Sqrt[3]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 2 k] 3^k a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}]

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * 3^k * a(n-2*k-1).
a(n) = Sum_{k=0..n} 3^((n-k)/2) * A136630(n,k). - Seiichi Manyama, Feb 20 2025

A381180 E.g.f. A(x) satisfies A(x) = 1 + sin(x*A(x)) / A(x).

Original entry on oeis.org

1, 1, 0, -1, -8, -19, 64, 1091, 7680, -1415, -650752, -8575865, -35559424, 857890021, 21380186112, 203548592651, -1615715926016, -95486152906639, -1599622990659584, -1397194164399601, 657963431581974528, 18168041375501245021, 157453907927886725120, -6059840564222790027821
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Crossrefs

Cf. A201627, A381181, A381182.
Cf. A136630, A196776.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n-k+1, k)/(n-k+1)*I^(n-k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(n-k+1,k)/(n-k+1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A381182 E.g.f. A(x) satisfies A(x) = 1/( 1 - A(x) * sin(x * A(x)) ).

Original entry on oeis.org

1, 1, 6, 71, 1288, 31661, 984640, 37085075, 1641305472, 83497838425, 4801347029504, 307975150996831, 21802395720298496, 1688562016007776261, 142023935786330431488, 12892154760586821775019, 1256251152910271399624704, 130793914073764385411654321, 14490427167940362294881615872
Offset: 0

Views

Author

Seiichi Manyama, Feb 16 2025

Keywords

Crossrefs

Cf. A201627, A381180, A381181.
Cf. A136630, A196776.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(n+2*k+1, k)/(n+2*k+1)*I^(n-k)*a136630(n, k));

Formula

a(n) = Sum_{k=0..n} k! * binomial(n+2*k+1,k)/(n+2*k+1) * i^(n-k) * A136630(n,k), where i is the imaginary unit.

A381278 Expansion of e.g.f. exp(sin(3*x) / 3).

Original entry on oeis.org

1, 1, 1, -8, -35, -8, 1117, 6328, -19943, -513728, -2096711, 30574720, 447401845, 23791744, -59033858219, -527680180736, 4971322421425, 144677554315264, 430091284739185, -27641200139694080, -398305237630617971, 2876369985206861824, 145441158283475935309
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Crossrefs

Cf. A002017, A009210.
Cf. A136630, A352640, A381286.

Programs

  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, (3*I)^(n-k)*a136630(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-9)^k * binomial(n-1,2*k) * a(n-2*k-1).
a(n) = Sum_{k=0..n} (3*i)^(n-k) * A136630(n,k), where i is the imaginary unit.
Previous Showing 41-50 of 88 results. Next

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