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 1-4 of 4 results.

A051163 Sequence is defined by property that (a0,a1,a2,a3,...) = binomial transform of (a0,a0,a1,a1,a2,a2,a3,a3,...).

Original entry on oeis.org

1, 2, 5, 12, 30, 76, 194, 496, 1269, 3250, 8337, 21428, 55184, 142376, 367916, 952000, 2466014, 6393372, 16586678, 43054344, 111801908, 290412296, 754543052, 1960808160, 5096293794, 13247503540, 34440553562, 89549255592, 232868582328, 605646682144
Offset: 0

Views

Author

Jonas Wallgren

Keywords

Comments

Equals the self-convolution of A027826. Also equals antidiagonal sums of symmetric square array A100936. - Paul D. Hanna, Nov 22 2004
Equals eigensequence of triangle A152198. - Gary W. Adamson, Nov 28 2008

Links

Crossrefs

Cf. A051164, A051165, A051166, A027826, A100936, A100937, A152198.

Programs

  • Maple
    a:= proc(n) option remember; add(`if`(k<2, 1,
      a(iquo(k, 2)))*binomial(n, k), k=0..n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 08 2015
  • Mathematica
    a[n_] := a[n] = 1 + Sum[Binomial[k, j]*Binomial[n-k, j]*a[j], {k, 1, n}, {j, 0, n-k}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 11 2015 *)
  • PARI
    a(n)=1+sum(k=1,n,sum(j=0,n-k,binomial(k,j)*binomial(n-k,j)*a(j)))
  • PARI
    a(n)=local(A,m); if(n<0,0,m=1; A=1+O(x); while(m<=n,m*=2; A=subst(A,x,(x/(1-x))^2)/(1-x)); polcoeff(A^2,n))
for(n=0,40,print1(a(n),", ")) \\ Paul D. Hanna, Nov 22 2004

Formula

a(n) = 1 + Sum_{k=1..n} Sum_{j=0..n-k} C(k, j)*C(n-k, j)*a(j). - Paul D. Hanna, Nov 22 2004
G.f. A(x) satisfies: A(x) = A(x^2/(1-x)^2)/(1-x)^2 and A(x^2) = A(x/(1+x))/(1+x)^2. - Paul D. Hanna, Nov 22 2004
a(0) = 1; a(n) = Sum_{k=0..floor(n/2)} binomial(n+1,2*k+1) * a(k). - Ilya Gutkovskiy, Apr 07 2022

Extensions

More terms from Vladeta Jovovic, Jul 26 2002

A051164 Sequence is defined by property that (a0,a1,a2,a3,...) = binomial transform of (a0,a0,a0,a1,a1,a1,a2,a2,a2,a3,a3,a3,...).

Original entry on oeis.org

1, 2, 4, 9, 21, 48, 108, 243, 549, 1243, 2819, 6412, 14640, 33549, 77121, 177747, 410565, 949992, 2200978, 5103623, 11839783, 27471189, 63734823, 147831594, 342767586, 794413545, 1840338975, 4261443374, 9863627962, 22822212734, 52789053456, 122073285984
Offset: 0

Views

Author

Jonas Wallgren

Keywords

Links

Crossrefs

Cf. A051163, A051165, A051166.

Programs

  • Maple
    a:= proc(n) option remember; add(`if`(k<3, 1,
      a(iquo(k, 3)))*binomial(n, k), k=0..n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 08 2015
  • Mathematica
    a[n_] := a[n] = Sum[If[k<3, 1, a[Quotient[k, 3]]]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2017, after Alois P. Heinz *)

Extensions

More terms from Vladeta Jovovic, Jul 26 2002

A051166 Sequence is defined by property that binomial transform of (a0,a1,a2,a3,...) = (a0,a0,a0,a1,a1,a1,a2,a2,a2,a3,a3,a3,...).

Original entry on oeis.org

1, 0, 0, -1, 3, -6, 10, -15, 21, -29, 45, -90, 224, -609, 1677, -4559, 12135, -31542, 80086, -199035, 485469, -1165105, 2757369, -6446778, 14913052, -34175805, 77672325, -175228740, 392711166, -874901088, 1938704130, -4275110880, 9385473510, -20521355211, 44704157499, -97055415324
Offset: 0

Views

Author

Jonas Wallgren

Keywords

Links

Crossrefs

Cf. A051163, A051164, A051165.

Programs

  • Maple
    a:= proc(n) option remember; add(`if`(k<3, 1,
      a(iquo(k, 3)))*(-1)^(n-k)*binomial(n, k), k=0..n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Jul 08 2015
  • Mathematica
    a[n_] := a[n] = Sum[If[k<3, 1, a[Quotient[k, 3]]]*(-1)^(n-k)*Binomial[n, k], {k, 0, n}];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Apr 04 2017, after Alois P. Heinz *)

Extensions

More terms from Vladeta Jovovic, Jul 26 2002

A059864 a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

Original entry on oeis.org

1, 1, 1, 2, 12, 96, 1152, 16128, 290304, 6967296, 181149696, 5796790272, 208684449792, 7930009092096, 333060381868032, 15986898329665536, 863292509801938944, 48344380548908580864, 2997351594032332013568
Offset: 1

Views

Author

Labos Elemer, Feb 28 2001

Keywords

Comments

Such products arise in Hardy-Littlewood prime k-tuplet conjectural formulas.

References

  • Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
  • R. K. Guy, Unsolved Problems in Number Theory, A8, A1
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
  • G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954

Links

Crossrefs

Cf. A000847, A002110, A005867, A022008, A048298, A049296, A051160, A051161, A051162.
Cf. A051163, A051164, A051165, A051166, A051167, A051168, A059861, A059862, A059863.
Cf. A059864, A059865.

Programs

  • Magma
    [n le 3 select 1 else (&*[NthPrime(j) -5: j in [4..n]]): n in [1..30]]; // G. C. Greubel, Feb 02 2023
  • Mathematica
    Join[{1,1,1},FoldList[Times,Prime[Range[4,20]]-5]] (* Harvey P. Dale, Dec 29 2018 *)
  • PARI
    a(n) = prod(k=4, n, prime(k)-5); \\ Michel Marcus, Dec 12 2017
  • SageMath
    def A059864(n): return product(nth_prime(j) -5 for j in range(4,n+1))
[A059864(n) for n in range(1,31)] # G. C. Greubel, Feb 02 2023
Showing 1-4 of 4 results.

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