A010745 -id:A010745 - cp's OEIS frontend

A010739 Shifts 2 places left under inverse binomial transform.

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

N. J. A. Sloane, Jonas Wallgren

Alois P. Heinz, Table of n, a(n) for n = 0..650
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A010741, A010743, A010745, A010747.

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

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 *)

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

A010741 Shifts 3 places left under inverse binomial transform.

Original entry on oeis.org

1, 2, 4, 1, 1, 1, -6, 14, -25, 32, 6, -250, 1222, -4380, 13059, -31705, 48464, 76354, -1159911, 7041015, -33400183, 135931668, -473704510, 1277600695, -1233828142, -16196871172, 169736941512, -1156974034428, 6577630531262, -32839667759307, 142900400342885

Offset: 0

N. J. A. Sloane, Jonas Wallgren

sign

Alois P. Heinz, Table of n, a(n) for n = 0..704
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A010739, A010743, A010745, A010747.

a:= proc(n) option remember; (m-> `if`(m<0, 2^n,
      add(a(m-j)*binomial(m, j)*(-1)^j, j=0..m)))(n-3)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 02 2022

a[n_] := a[n] = With[{m = n - 3}, If[m < 0, 2^n,
     Sum[a[m - j]*Binomial[m, j]*(-1)^j, {j, 0, m}]]];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Mar 02 2022, after Alois P. Heinz *)

G.f. A(x) satisfies: A(x) = 1 + 2*x + 4*x^2 + x^3*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

N. J. A. Sloane, Jonas Wallgren

sign

Alois P. Heinz, Table of n, a(n) for n = 0..740
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A010739, A010741, A010745, A010747.

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

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 *)

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

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

N. J. A. Sloane, Jonas Wallgren

Alois P. Heinz, Table of n, a(n) for n = 0..802
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A010739, A010741, A010743, A010745.

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

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 *)

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

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A010739 Shifts 2 places left under inverse binomial transform.

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A010741 Shifts 3 places left under inverse binomial transform.

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A010743 Shifts 4 places left under inverse binomial transform.

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A010747 Shifts 6 places left under inverse binomial transform.

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Maple

Mathematica

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