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.

Showing 1-6 of 6 results.

A385835 a(n) = 1 + Sum_{k=0..n-1} (1 + k^2) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 7, 51, 660, 13350, 390886, 15728919, 836469748, 56989647229, 4849599126797, 504709937298467, 63117270187248665, 9344222191368190761, 1616899887657388367640, 323430766605746093449465, 74074314477265886578774322, 19261037812212680097678843345, 5643873902659784713257894768422
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Links

Crossrefs

Cf. A321087, A385836, A385837, A385838, A385839.
Cf. A385758.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, (1+j^2)*v[j+1]*v[i-j])); v;

Formula

G.f. A(x) satisfies A(x) = 1/( (1 - x) * ( 1 - x*A(x) - x^2 * (d/dx A(x)) - x^3 * (d^2/dx^2 A(x)) ) ).

A385759 G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x*A(x) - x^4*A'''(x))).

Original entry on oeis.org

1, 2, 5, 15, 141, 3932, 251717, 31216948, 6680698525, 2271470142438, 1153913665217481, 835435792656039975, 830424340158140342961, 1099482665756962845820704, 1891111018270919721409143729, 4137752010118540256190073466415, 11312615890237585633045672755792789
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A321087, A385758, A385760, A385761.

Programs

  • Mathematica
    terms = 17; A[] = 0; Do[A[x] = 1/((1-x)*(1-x*A[x]-x^4*A'''[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 09 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, (1+sum(k=1, 3, stirling(3, k, 1)*j^k))*v[j+1]*v[i-j])); v;

Formula

a(n) = 1 + Sum_{k=0..n-1} (1 + 2*k - 3*k^2 + k^3) * a(k) * a(n-1-k).

A385760 G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x*A(x) - x^5*A''''(x))).

Original entry on oeis.org

1, 2, 5, 15, 51, 1412, 175067, 63725638, 53784616915, 90573359145678, 274256185472187231, 1383348290257488337035, 10961652126528967555229301, 130268275255842369871718355444, 2235924687457083597476492688851325, 53724798520519979444347750309693062183
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A321087, A385758, A385759, A385761.

Programs

  • Mathematica
    terms = 16; A[] = 0; Do[A[x] = 1/((1-x)*(1-x*A[x]-x^5*A''''[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 09 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, (1+sum(k=1, 4, stirling(4, k, 1)*j^k))*v[j+1]*v[i-j])); v;

Formula

a(n) = 1 + Sum_{k=0..n-1} (1 - 6*k + 11*k^2 - 6*k^3 + k^4) * a(k) * a(n-1-k).

A385761 G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x*A(x) - x^6*A'''''(x))).

Original entry on oeis.org

1, 2, 5, 15, 51, 188, 23291, 16862710, 42561503035, 286183563337662, 4328240254531111671, 130903298544350358627387, 7257802488822060515691899445, 689810579878520205782663179307100, 106537105206016369903910237449838232525, 25594900303804029125790200935921438169789415
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A321087, A385758, A385759, A385760.

Programs

  • Mathematica
    terms = 16; A[] = 0; Do[A[x] = 1/((1-x)*(1-x*A[x]-x^6*A'''''[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jul 09 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, (1+sum(k=1, 5, stirling(5, k, 1)*j^k))*v[j+1]*v[i-j])); v;

Formula

a(n) = 1 + Sum_{k=0..n-1} (1 + 24*k - 50*k^2 + 35*k^3 - 10*k^4 + k^5) * a(k) * a(n-1-k).

A385844 G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x^3*A''(x))).

Original entry on oeis.org

1, 1, 1, 3, 21, 273, 5737, 177919, 7651849, 436186313, 31842549569, 2897710853939, 321648004495773, 42779331295225353, 6716367934603667145, 1229096733282700520799, 259339594018913458094865, 62500870590534491566841265, 17062742827503910747790541249, 5238263128497776755775631825219
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2025

Keywords

Crossrefs

Cf. A143917, A385845, A385846.
Cf. A385758, A385762.

Programs

  • Mathematica
    terms = 20; A[] = 0; Do[A[x] = 1/((1 - x) * (1 - x^3*A''[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]  (* Stefano Spezia, Jul 10 2025 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, sum(k=1, 2, stirling(2, k, 1)*j^k)*v[j+1]*v[i-j])); v;

Formula

a(n) = 1 + Sum_{k=0..n-1} (-k + k^2) * a(k) * a(n-1-k).

A386071 Primes having only {0, 4, 5, 9} as digits.

Original entry on oeis.org

5, 59, 409, 449, 499, 509, 599, 4049, 4099, 4409, 4549, 4909, 4999, 5009, 5059, 5099, 5449, 9049, 9059, 9949, 40009, 40099, 40459, 40499, 40559, 40949, 44059, 44449, 44549, 44909, 44959, 45599, 45949, 45959, 49009, 49409, 49459, 49499, 49549, 49559, 49999, 50459
Offset: 1

Views

Author

Jason Bard, Jul 16 2025

Keywords

Links

Crossrefs

Supersequence of A385758, A385769, A385793.
Cf. A000040, A030433, A385776.

Programs

  • Magma
    [p: p in PrimesUpTo(10^6) | Set(Intseq(p)) subset [0, 4, 5, 9]];
  • Mathematica
    Select[FromDigits /@ Tuples[{0, 4, 5, 9}, n], PrimeQ]
  • PARI
    primes_with(, 1, [0, 4, 5, 9]) \\ uses function in A385776
  • Python
    print(list(islice(primes_with("0459"), 41))) # uses function/imports in A385776
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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