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.

A007472 Shifts 2 places left when binomial transform is applied twice with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 1, 3, 9, 29, 105, 431, 1969, 9785, 52145, 296155, 1787385, 11428949, 77124569, 546987143, 4062341601, 31502219889, 254500383457, 2137863653811, 18639586581097, 168387382189709, 1573599537048265, 15189509662516063, 151243491212611217, 1551565158004180137
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A351028, A351143.
Row sums of triangle A383235.

Programs

  • Maple
    bintr:= proc(p) local b;
          b:= proc(n) option remember; add(p(k)*binomial(n,k), k=0..n) end
        end:
b:= (bintr@@2)(a):
a:= n-> `if`(n<2, 1, b(n-2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 18 2012
  • Mathematica
    bintr[p_] := Module[{b}, b[n_] := b[n] = Sum [p[k]*Binomial[n, k], {k, 0, n}]; b]; b = a // bintr // bintr; a[n_] := If[n<2, 1, b[n-2]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 27 2014, after Alois P. Heinz *)
(* another program *)
B[x_] := (BesselK[0, 1] + BesselK[1, 1])*BesselI[0, Exp[x]] + (BesselI[1, 1] - BesselI[0, 1])*BesselK[0, Exp[x]];
a[n_] := SeriesCoefficient[FullSimplify[Series[B[x], {x, 0, n}]],n] n!
Table[a[n], {n, 0, 30}] (* Ven Popov, Apr 25 2025 *)

Formula

G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 2*x)) / (1 - 2*x). - Ilya Gutkovskiy, Jan 30 2022
E.g.f.: (BesselK(0, 1) + BesselK(1, 1)) * BesselI(0, exp(x)) + (BesselI(1, 1) - BesselI(0, 1)) * BesselK(0, exp(x)). - Ven Popov, Apr 25 2025

A351028 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 2*x)) / (1 - 2*x).

Original entry on oeis.org

0, 1, 0, 1, 4, 13, 44, 173, 792, 4009, 21608, 122761, 737340, 4696341, 31665076, 224846037, 1672266352, 12976252561, 104816144656, 880061135057, 7670326372532, 69286959112797, 647568753568636, 6251768635591613, 62255057942504968, 638658964709824185
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 03 2022

Keywords

Comments

Shifts 2 places left under 2nd-order binomial transform.

Crossrefs

Cf. A000995, A004211, A007472, A351053, A351128, A351132, A351143, A351161.

Programs

  • Maple
    bintr:= proc(p) local b;
          b:= proc(n) option remember; add(p(k)*binomial(n, k), k=0..n) end
        end:
b:= (bintr@@2)(a):
a:= n-> `if`(n<2, n, b(n-2)):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 07 2025
  • Mathematica
    nmax = 25; A[] = 0; Do[A[x] = x + x^2 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] 2^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}];
(* another pprogram *)
B[x_] := BesselK[0, 1]*BesselI[0, Exp[x]] - BesselI[0, 1]*BesselK[0, Exp[x]];
a[n_] := SeriesCoefficient[FullSimplify[Series[B[x], {x, 0, n}]], n] n!;
Table[a[n], {n, 0, 30}] (* Ven Popov, Apr 25 2025 *)

Formula

a(0) = 0, a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 2^k * a(n-k-2).
E.g.f.: BesselK(0, 1)*BesselI(0, exp(x)) - BesselI(0, 1)*BesselK(0, exp(x)). - Ven Popov, Apr 25 2025

A351144 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 3*x)) / (1 - 3*x).

Original entry on oeis.org

1, 0, 1, 3, 10, 39, 181, 972, 5797, 37389, 258202, 1905681, 15016465, 125920872, 1117950913, 10452866439, 102485649754, 1050464300187, 11231883627301, 125055844922916, 1447371528438565, 17382103226123313, 216221862096537994, 2781342531957176085, 36942930754308211969
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 02 2022

Keywords

Comments

Shifts 2 places left under 3rd-order binomial transform.

Crossrefs

Cf. A000994, A004212, A351049, A351143, A351150, A351151, A351152.

Programs

  • Mathematica
    nmax = 24; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 3 x)]/(1 - 3 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 3^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 24}]

Formula

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

A351150 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 4*x)) / (1 - 4*x).

Original entry on oeis.org

1, 0, 1, 4, 17, 80, 433, 2724, 19489, 153536, 1301601, 11754084, 112802097, 1150079056, 12437130001, 142144768324, 1709041379393, 21522252928000, 282920962675905, 3873124754702660, 55125031662585425, 814541756986322128, 12477752083406752881, 197861190429889969252
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 02 2022

Keywords

Comments

Shifts 2 places left under 4th-order binomial transform.

Crossrefs

Cf. A000994, A004213, A351050, A351143, A351144, A351151, A351152.

Programs

  • Mathematica
    nmax = 23; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 4 x)]/(1 - 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 4^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 23}]

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 4^k * a(n-k-2).

A351151 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 5*x)) / (1 - 5*x).

Original entry on oeis.org

1, 0, 1, 5, 26, 145, 901, 6420, 52501, 480955, 4795626, 51066375, 576182001, 6879462680, 86955722401, 1162559359745, 16392133866026, 242734091500445, 3758825675820501, 60660434188558780, 1017770666417312501, 17725289455315892375, 320047193447632729626
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 02 2022

Keywords

Comments

Shifts 2 places left under 5th-order binomial transform.

Crossrefs

Cf. A000994, A005011, A351056, A351143, A351144, A351150, A351152.

Programs

  • Mathematica
    nmax = 22; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 5 x)]/(1 - 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 5^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 5^k * a(n-k-2).

A351152 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 6*x)) / (1 - 6*x).

Original entry on oeis.org

1, 0, 1, 6, 37, 240, 1693, 13446, 122329, 1261104, 14332681, 175123446, 2267871517, 30981705984, 446571784261, 6798161166486, 109220619908593, 1846729159654560, 32726973173941585, 605358657750562470, 11648701234354836565, 232655173657593759312
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 02 2022

Keywords

Comments

Shifts 2 places left under 6th-order binomial transform.

Crossrefs

Cf. A000994, A005012, A351057, A351143, A351144, A351150, A351151.

Programs

  • Mathematica
    nmax = 21; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 6 x)]/(1 - 6 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 0; a[n_] := a[n] = Sum[Binomial[n - 2, k] 6^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 21}]

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 6^k * a(n-k-2).
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.