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.

A010739 Shifts 2 places left under inverse binomial transform.

Original entry on oeis.org

1, 2, 1, 1, -2, 3, -7, 22, -71, 231, -794, 2945, -11679, 48770, -212823, 969221, -4605674, 22802431, -117322423, 625743878, -3452893503, 19684083947, -115787084242, 701935339725, -4380330298815, 28105726916034, -185229395693615, 1252696143653513
Offset: 0

Views

Author

N. J. A. Sloane, Jonas Wallgren

Keywords

Links

Crossrefs

Cf. A010741, A010743, A010745, A010747.

Programs

  • Maple
    a:= proc(n) option remember; (m-> `if`(m<0, 2^n,
      add(a(m-j)*binomial(m, j)*(-1)^j, j=0..m)))(n-2)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 02 2022
  • Mathematica
    a[n_] := a[n] = Function[m, If[m<0, 2^n,
   Sum[a[m-j]*Binomial[m, j]*(-1)^j, {j, 0, m}]]][n-2];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jul 24 2022, after Alois P. Heinz *)

Formula

G.f. A(x) satisfies: A(x) = 1 + 2*x + x^2*A(x/(1 + x))/(1 + x). - Ilya Gutkovskiy, Feb 02 2022

A010743 Shifts 4 places left under inverse binomial transform.

Original entry on oeis.org

1, 2, 4, 8, 1, 1, 1, 1, -14, 45, -101, 189, -331, 668, -1932, 7206, -27779, 101365, -347439, 1139851, -3690766, 12258863, -43341845, 166059261, -682516519, 2930522990, -12823188092, 56366526324, -247898684759, 1094571175769, -4890163717903, 22310147976797
Offset: 0

Views

Author

N. J. A. Sloane, Jonas Wallgren

Keywords

Links

Crossrefs

Cf. A010739, A010741, A010745, A010747.

Programs

  • Maple
    a:= proc(n) option remember; (m-> `if`(m<0, 2^n,
      add(a(m-j)*binomial(m, j)*(-1)^j, j=0..m)))(n-4)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 02 2022
  • Mathematica
    a[n_] := a[n] = Function[m, If[m < 0, 2^n, Sum[a[m - j]* Binomial[m, j]*(-1)^j, {j, 0, m}]]][n - 4];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Mar 08 2023, after Alois P. Heinz *)

Formula

G.f. A(x) satisfies: A(x) = 1 + 2*x + 4*x^2 + 8*x^3 + x^4*A(x/(1 + x))/(1 + x). - Ilya Gutkovskiy, Feb 02 2022

A010745 Shifts 5 places left under inverse binomial transform.

Original entry on oeis.org

1, 2, 4, 8, 16, 1, 1, 1, 1, 1, -30, 124, -336, 734, -1401, 2404, -3485, 2212, 14630, -105408, 497131, -1995782, 7265342, -24576128, 77966104, -231218343, 626012198, -1430352680, 1894959964, 6114950887, -73791743479, 472896657475, -2523776826105, 12272646042530
Offset: 0

Views

Author

N. J. A. Sloane, Jonas Wallgren

Keywords

Links

Crossrefs

Cf. A010739, A010741, A010743, A010747.

Programs

  • Maple
    a:= proc(n) option remember; (m-> `if`(m<0, 2^n,
      add(a(m-j)*binomial(m, j)*(-1)^j, j=0..m)))(n-5)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 02 2022
  • Mathematica
    a[n_] := a[n] = Function[m, If[m < 0, 2^n, Sum[a[m-j]*Binomial[m, j]*(-1)^j, {j, 0, m}]]][n-5]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jan 13 2025, after Alois P. Heinz *)

Formula

G.f. A(x) satisfies: A(x) = 1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + x^5*A(x/(1 + x))/(1 + x). - Ilya Gutkovskiy, Feb 02 2022

A010747 Shifts 6 places left under inverse binomial transform.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 1, 1, 1, 1, 1, 1, -62, 315, -1002, 2505, -5377, 10373, -18544, 32086, -60468, 154687, -567986, 2422043, -10225382, 40740231, -152497274, 539809668, -1822828757, 5928049329, -18782176673, 58918636670, -187382010256, 623524250516
Offset: 0

Views

Author

N. J. A. Sloane, Jonas Wallgren

Keywords

Links

Crossrefs

Cf. A010739, A010741, A010743, A010745.

Programs

  • Maple
    a:= proc(n) option remember; (m-> `if`(m<0, 2^n,
      add(a(m-j)*binomial(m, j)*(-1)^j, j=0..m)))(n-6)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 02 2022
  • Mathematica
    a[n_] := a[n] = Function[m, If[m < 0, 2^n, Sum[a[m - j]* Binomial[m, j]*(-1)^j, {j, 0, m}]]][n - 6];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Mar 08 2023, after Alois P. Heinz *)

Formula

G.f. A(x) satisfies: A(x) = 1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + 32*x^5 + x^6*A(x/(1 + x))/(1 + x). - Ilya Gutkovskiy, Feb 02 2022
a(n) = 2^n for n<6; otherwise a(n) = Sum_{j=0..n-6} a(n-6-j)*binomial(n-6,j)*(-1)^j. - Michel Marcus, Mar 08 2023

A351188 G.f. A(x) satisfies: A(x) = x + x^3 * A(x/(1 + x)) / (1 + x).

Original entry on oeis.org

0, 1, 0, 0, 1, -2, 3, -3, -2, 24, -94, 280, -687, 1270, -655, -9306, 65087, -306724, 1202250, -4033365, 10855578, -15470865, -69819687, 850568716, -5679272040, 30749200898, -144912453016, 593521998765, -1956641103972, 3349999455415, 20123433921282, -295760693980981
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2022

Keywords

Comments

Shifts 3 places left under inverse binomial transform.

Crossrefs

Cf. A000997, A010741, A351189.

Programs

  • Mathematica
    nmax = 31; A[] = 0; Do[A[x] = x + x^3 A[x/(1 + x)]/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 0; a[1] = 1; a[2] = 0; a[n_] := a[n] = Sum[(-1)^k Binomial[n - 3, k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]

Formula

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

A351189 G.f. A(x) satisfies: A(x) = x^2 + x^3 * A(x/(1 + x)) / (1 + x).

Original entry on oeis.org

0, 0, 1, 0, 0, 1, -3, 6, -9, 6, 27, -169, 645, -1995, 5122, -9570, 1242, 109739, -756648, 3733128, -15527337, 55626585, -161247102, 260402511, 1028417064, -14243992155, 102551438561, -595149283191, 3010031905815, -13336771020834, 48891499316016, -111677138548476
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2022

Keywords

Comments

Shifts 3 places left under inverse binomial transform.

Crossrefs

Cf. A000998, A010741, A351188.

Programs

  • Mathematica
    nmax = 31; A[] = 0; Do[A[x] = x^2 + x^3 A[x/(1 + x)]/(1 + x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 0; a[2] = 1; a[n_] := a[n] = Sum[(-1)^k Binomial[n - 3, k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 31}]

Formula

a(0) = a(1) = 0, a(2) = 1; a(n) = Sum_{k=0..n-3} (-1)^k * binomial(n-3,k) * a(n-k-3).
Showing 1-6 of 6 results.

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

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