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.

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).

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).

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).

A351128 G.f. A(x) satisfies: A(x) = x + x^2 * A(x/(1 - 4*x)) / (1 - 4*x).

Original entry on oeis.org

0, 1, 0, 1, 8, 49, 280, 1649, 10800, 81505, 696400, 6472033, 63562872, 652984977, 7026210728, 79547049681, 949709767904, 11936248012993, 157219119485216, 2159448120457409, 30811324011852136, 455635009201780977, 6975424580445456056, 110478282815356437809
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 03 2022

Keywords

Comments

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

Crossrefs

Cf. A000995, A004213, A351028, A351050, A351053, A351132, A351150, A351161.

Programs

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

Formula

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

A351185 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, -3, 9, -31, 153, -1075, 8689, -72031, 605201, -5282051, 49239225, -497094079, 5410919273, -62597718643, 759331611489, -9586004915007, 125701843190689, -1713676634245251, 24313707650733289, -358906747784541151, 5502327502961296825, -87382907614533531443
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 04 2022

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

a(0) = a(1) = 1; a(n) = Sum_{k=0..n-2} binomial(n-2,k) * (-4)^k * a(n-k-2).
Showing 1-6 of 6 results.

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