A010739 -id:A010739 - cp's OEIS frontend

A010741 Shifts 3 places left under inverse binomial transform.

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

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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

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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

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

N. J. A. Sloane, Jonas Wallgren

sign

Alois P. Heinz, Table of n, a(n) for n = 0..775
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 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, A010743, 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-5)
    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-5]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jan 13 2025, after Alois P. Heinz *)

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

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

A350456 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 2x)) / (1 + 2x).

Original entry on oeis.org

1, 1, 1, -1, 1, -3, 17, -85, 385, -1767, 8929, -50633, 312705, -2036267, 13794417, -97295069, 717808897, -5549714767, 44868094145, -377741383697, 3298933836033, -29813463964115, 278462029910993, -2685972391332837, 26733375327601281, -274247228584531767

Offset: 0

Ilya Gutkovskiy, Feb 04 2022

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

Cf. A007472, A009235, A010739, A051139, A351184, A351185, A351186, A351187.

nmax = 25; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 + 2 x)]/(1 + 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[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}]

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

A351184 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 3x)) / (1 + 3x).

Original entry on oeis.org

1, 1, 1, -2, 4, -11, 55, -359, 2359, -15230, 100840, -716555, 5580145, -47230091, 425472229, -4013326982, 39379161136, -402010392971, 4279164575167, -47533936734179, 550239127112107, -6618018093867506, 82447377648018700, -1061324336149876667, 14095604842846277617

Offset: 0

Ilya Gutkovskiy, Feb 04 2022

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

Cf. A010739, A051139, A317996, A350456, A351049, A351185, A351186, A351187.

nmax = 24; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 + 3 x)]/(1 + 3 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; 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}]

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

A351185 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 4x)) / (1 + 4x).

Original entry on oeis.org

1, 1, 1, -3, 9, -31, 153, -1075, 8689, -72031, 605201, -5282051, 49239225, -497094079, 5410919273, -62597718643, 759331611489, -9586004915007, 125701843190689, -1713676634245251, 24313707650733289, -358906747784541151, 5502327502961296825, -87382907614533531443

Offset: 0

Ilya Gutkovskiy, Feb 04 2022

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

Cf. A010739, A051139, A318179, A350456, A351050, A351184, A351186, A351187.

nmax = 23; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 + 4 x)]/(1 + 4 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; 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}]

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

A351186 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 5x)) / (1 + 5x).

Original entry on oeis.org

1, 1, 1, -4, 16, -69, 371, -2719, 24691, -243804, 2479276, -25931249, 284075601, -3320433179, 41744590941, -561939568544, 8008026088996, -119496752915869, 1854697111334891, -29870689367146379, 499291484226079551, -8668202648905259624, 156301404533216141576

Offset: 0

Ilya Gutkovskiy, Feb 04 2022

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

Cf. A010739, A051139, A318180, A350456, A351056, A351184, A351185, A351187.

nmax = 22; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 + 5 x)]/(1 + 5 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; 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}]

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

A351187 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 6x)) / (1 + 6x).

Original entry on oeis.org

1, 1, 1, -5, 25, -131, 793, -6137, 60049, -670919, 7930321, -96775853, 1225237609, -16333089227, 232150489129, -3531321746465, 57178717416097, -975918663642767, 17400776511175201, -322309002081819221, 6188520430773389881, -123166171374344928275, 2542231599282355411897

Offset: 0

Ilya Gutkovskiy, Feb 04 2022

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

Cf. A010739, A051139, A318181, A350456, A351057, A351184, A351185, A351186.

nmax = 22; A[] = 0; Do[A[x] = 1 + x + x^2 A[x/(1 + 6 x)]/(1 + 6 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n - 2, k] (-6)^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 22}]

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

A379781 a(1)=1, a(2)=2; thereafter, a(n) is the final value at the bottom of the difference triangle of the sequence thus far.

Original entry on oeis.org

1, 2, 1, -2, 0, 7, -14, -12, 155, -408, -364, 7693, -30940, 10712, 637701, -4224222, 9980180, 61922567, -810337234, 4100137008, -958593005, -174952472228, 1662063951016, -6944673371867, -22887336602200, 655644589917172, -5694691183524699, 19946666531550638, 176993602416669640

Offset: 1

Neal Gersh Tolunsky, Jan 02 2025

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The difference triangle refers to the triangular array of iterated differences.

The first term in each row of the difference triangle is the inverse binomial transform of the sequence, so the definition means the inverse binomial transform deletes term a(2) = 2.

			To find a(6), we look at the first difference triangle of the first 5 terms:
  1,   2,   1,  -2,   0
  1,  -1,  -3,   2
 -2,  -2,   5
  0,   7
  7
7 is the final value, so a(6)=7.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..595

Cf. A014182, A010739.

a[1] = 1; a[2] = 2; a[n_] := a[n] = Sum[(-1)^(n-k+1) * Binomial[n-2, k-1] * a[k], {k, 1, n-1}]; Table[a[n], {n, 1, 30}] (* Amiram Eldar, Jan 04 2025 *)

cp's OEIS Frontend

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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A010745 Shifts 5 places left under inverse binomial transform.

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A010747 Shifts 6 places left under inverse binomial transform.

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A350456 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 2*x)) / (1 + 2*x).

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A351184 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 3*x)) / (1 + 3*x).

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A351185 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 4*x)) / (1 + 4*x).

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A351186 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 5*x)) / (1 + 5*x).

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A351187 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 6*x)) / (1 + 6*x).

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A350456 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 2x)) / (1 + 2x).

A351184 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 3x)) / (1 + 3x).

A351185 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 4x)) / (1 + 4x).

A351186 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 5x)) / (1 + 5x).

A351187 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 + 6x)) / (1 + 6x).