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 1-10 of 12 results. Next

A351049 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 3*x)) / (1 - 3*x).

Original entry on oeis.org

1, 1, 1, 4, 16, 67, 307, 1585, 9235, 59548, 415564, 3094807, 24452785, 204611653, 1810429597, 16892405896, 165592138372, 1698918207403, 18184602679435, 202577753111653, 2344503929765023, 28146188358379120, 349996346545057288, 4501360727764475503
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2022

Keywords

Comments

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

Links

Crossrefs

Cf. A004212, A007472, A007476, A351050.

Programs

  • Mathematica
    nmax = 23; 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, 23}]

Formula

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

A351050 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 4*x)) / (1 - 4*x).

Original entry on oeis.org

1, 1, 1, 5, 25, 129, 713, 4373, 30289, 235041, 1998001, 18226117, 176364969, 1803064033, 19463340729, 221691818005, 2658751147297, 33458500940993, 440140082161121, 6032572875160069, 85936355674437561, 1270176766188103105, 19453176663852208937
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2022

Keywords

Comments

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

Links

Crossrefs

Cf. A004213, A007472, A007476, A351049.

Programs

  • Mathematica
    nmax = 22; 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, 22}]

Formula

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

A351143 G.f. A(x) satisfies: A(x) = 1 + x^2 * A(x/(1 - 2*x)) / (1 - 2*x).

Original entry on oeis.org

1, 0, 1, 2, 5, 16, 61, 258, 1177, 5776, 30537, 173394, 1050045, 6732608, 45459493, 322141106, 2390075249, 18525967328, 149684238801, 1257802518754, 10969260208565, 99100423076912, 926030783479629, 8937741026924450, 88988433270106249, 912906193294355952
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 02 2022

Keywords

Comments

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

Crossrefs

Cf. A000994, A004211, A007472, A351144, A351150, A351151, A351152.

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-n, b(n-2)):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 07 2025
  • Mathematica
    nmax = 25; A[] = 0; Do[A[x] = 1 + x^2 A[x/(1 - 2 x)]/(1 - 2 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] 2^k a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 25}]
(* another program *)
B[x_] := BesselK[1, 1]*BesselI[0, Exp[x]] + BesselI[1, 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) = 1, a(1) = 0; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * 2^k * a(n-k-2).
E.g.f.: BesselK(1, 1)*BesselI(0, exp(x)) + BesselI(1, 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

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

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

Views

Author

Ilya Gutkovskiy, Feb 04 2022

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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

A351056 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 5*x)) / (1 - 5*x).

Original entry on oeis.org

1, 1, 1, 6, 36, 221, 1431, 10121, 80311, 718106, 7111976, 76201501, 868288401, 10438492181, 132166853861, 1763179150946, 24776241643056, 365971430085021, 5662954240306111, 91450179009971181, 1536249848608545451, 26782376261726525126, 483792982362049317676
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2022

Keywords

Comments

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

Crossrefs

Cf. A005011, A007472, A007476, A351049, A351050, A351057.

Programs

  • Mathematica
    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}]

Formula

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

A351057 G.f. A(x) satisfies: A(x) = 1 + x + x^2 * A(x/(1 - 6*x)) / (1 - 6*x).

Original entry on oeis.org

1, 1, 1, 7, 49, 349, 2593, 20755, 184609, 1851289, 20735041, 253471039, 3310505425, 45630322741, 660993079393, 10065000586507, 161262522401089, 2717539655666353, 48053169836707969, 888408313419305719, 17108882037936283249, 342144175940842590349, 7089944927940141776545
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 30 2022

Keywords

Comments

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

Crossrefs

Cf. A005012, A007472, A007476, A351049, A351050, A351056.

Programs

  • Mathematica
    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}]

Formula

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

A351342 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x/(1 - 2*x)) / (1 - 2*x).

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 27, 83, 271, 971, 3865, 16879, 78985, 388385, 1987201, 10561385, 58443891, 337724057, 2040085491, 12862712499, 84357800063, 573182197539, 4021203303593, 29062345301487, 216129411635057, 1653180368063361, 13003920016983361, 105158133803473329
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 08 2022

Keywords

Comments

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

Crossrefs

Cf. A000996, A000998, A004211, A007472, A210540, A351343, A351344, A351345.

Programs

  • Mathematica
    nmax = 27; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 3, 1, Sum[Binomial[n - 3, k] 2^k a[n - k - 3], {k, 0, n - 3}]]; Table[a[n], {n, 0, 27}]

Formula

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

A351343 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 * A(x/(1 - 2*x)) / (1 - 2*x).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 9, 27, 81, 245, 761, 2493, 8849, 34519, 147057, 670327, 3198561, 15732905, 79174929, 407127897, 2145061729, 11635963499, 65309080185, 380583443187, 2304629301041, 14475031232285, 93943897651017, 627220447621973, 4290783719133041, 29988917377046207
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 08 2022

Keywords

Comments

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

Crossrefs

Cf. A004211, A007472, A010748, A210541, A275934, A351342, A351344, A351345.

Programs

  • Mathematica
    nmax = 29; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 4, 1, Sum[Binomial[n - 4, k] 2^k a[n - k - 4], {k, 0, n - 4}]]; Table[a[n], {n, 0, 29}]

Formula

a(0) = ... = a(3) = 1; a(n) = Sum_{k=0..n-4} binomial(n-4,k) * 2^k * a(n-k-4).

A351344 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4 + x^5 * A(x/(1 - 2*x)) / (1 - 2*x).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 9, 27, 81, 243, 731, 2223, 6939, 22727, 79971, 306929, 1282815, 5744361, 26984415, 130656409, 644739377, 3224303841, 16318576681, 83717193681, 436948772697, 2331807007139, 12791837178265, 72472130039123, 425239734375217, 2584950704996379
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 08 2022

Keywords

Comments

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

Crossrefs

Cf. A004211, A007472, A010749, A210542, A275935, A351342, A351343, A351345.

Programs

  • Mathematica
    nmax = 30; A[] = 0; Do[A[x] = 1 + x + x^2 + x^3 + x^4 + x^5 A[x/(1 - 2 x)]/(1 - 2 x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = If[n < 5, 1, Sum[Binomial[n - 5, k] 2^k a[n - k - 5], {k, 0, n - 5}]]; Table[a[n], {n, 0, 30}]

Formula

a(0) = ... = a(4) = 1; a(n) = Sum_{k=0..n-5} binomial(n-5,k) * 2^k * a(n-k-5).
Showing 1-10 of 12 results. Next

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