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 1-5 of 5 results.

A290890 p-INVERT of the positive integers, where p(S) = 1 - S^2.

Original entry on oeis.org

0, 1, 4, 11, 28, 72, 188, 493, 1292, 3383, 8856, 23184, 60696, 158905, 416020, 1089155, 2851444, 7465176, 19544084, 51167077, 133957148, 350704367, 918155952, 2403763488, 6293134512, 16475640049, 43133785636, 112925716859, 295643364940, 774004377960
Offset: 0

Views

Author

Clark Kimberling, Aug 15 2017

Keywords

Comments

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
Note that in A290890, s = (1,2,3,4,...); i.e., A000027(n+1) for n>=0, whereas in A290990, s = (0,1,2,3,4,...); i.e., A000027(n) for n>=0.
Guide to p-INVERT sequences using s = (1,2,3,4,5,...) = A000027:
p(S) t(1,2,3,4,5,...)
1 - S A001906
1 - S^2 A290890; see A113067 for signed version
1 - S^3 A290891
1 - S^4 A290892
1 - S^5 A290893
1 - S^6 A290894
1 - S^7 A290895
1 - S^8 A290896
1 - S - S^2 A289780
1 - S - S^3 A290897
1 - S - S^4 A290898
1 - S^2 - S^4 A290899
1 - S^2 - S^3 A290900
1 - S^3 - S^4 A290901
1 - 2S A052530; (1/2)*A052530 = A001353
1 - 3S A290902; (1/3)*A290902 = A004254
1 - 4S A003319; (1/4)*A003319 = A001109
1 - 5S A290903; (1/5)*A290903 = A004187
1 - 2*S^2 A290904; (1/2)*A290904 = A290905
1 - 3*S^2 A290906; (1/3)*A290906 = A290907
1 - 4*S^2 A290908; (1/4)*A290908 = A099486
1 - 5*S^2 A290909; (1/5)*A290909 = A290910
1 - 6*S^2 A290911; (1/6)*A290911 = A290912
1 - 7*S^2 A290913; (1/7)*A290913 = A290914
1 - 8*S^2 A290915; (1/8)*A290915 = A290916
(1 - S)^2 A290917
(1 - S)^3 A290918
(1 - S)^4 A290919
(1 - S)^5 A290920
(1 - S)^6 A290921
1 - S - 2*S^2 A290922
1 - 2*S - 2*S^2 A290923; (1/2)*A290923 = A290924
1 - 3*S - 2*S^2 A290925
(1 - S^2)^2 A290926
(1 - S^2)^3 A290927
(1 - S^3)^2 A290928
(1 - S)(1 - S^2) A290929
(1 - S^2)(1 - S^4) A290930
1 - 3 S + S^2 A291025
1 - 4 S + S^2 A291026
1 - 5 S + S^2 A291027
1 - 6 S + S^2 A291028
1 - S - S^2 - S^3 A291029
1 - S - S^2 - S^3 - S^4 A201030
1 - 3 S + 2 S^3 A291031
1 - S - S^2 - S^3 + S^4 A291032
1 - 6 S A291033
1 - 7 S A291034
1 - 8 S A291181
1 - 3 S + 2 S^3 A291031
1 - 3 S + 2 S^2 A291182
1 - 4 S + 2 S^3 A291183
1 - 4 S + 3 S^3 A291184

Examples

			(See the examples at A289780.)

Links

Crossrefs

Cf. A000027, A113067, A289780, A113067 (signed version of same sequence).

Programs

  • Mathematica
    z = 60; s = x/(1 - x)^2; p = 1 - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290890 *)

Formula

G.f.: x/(1 - 4 x + 5 x^2 - 4 x^3 + x^4).
a(n) = 4*a(n-1) - 5*a(n-2) + 4*a(n-3) - a(n-4).

A099487 Expansion of (1-3x+x^2)/((1+x^2)(1-4x+x^2)).

Original entry on oeis.org

1, 1, 3, 14, 53, 195, 727, 2716, 10137, 37829, 141179, 526890, 1966381, 7338631, 27388143, 102213944, 381467633, 1423656585, 5313158707, 19828978246, 74002754277, 276182038859, 1030725401159, 3846719565780, 14356152861961
Offset: 0

Views

Author

Paul Barry, Oct 18 2004

Keywords

Comments

A Chebyshev transform of the sequence 1,1,4,16.. which has with g.f. (1-3x)/(1-4x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).

Links

Crossrefs

Cf. A099486, A099488.

Formula

a(n)=4a(n-1)-2a(n-2)+4a(n-3); a(n)=sum{k=0..floor(n/2), C(n-k, k)(-1)^n*(4^(n-2k)+3*0^(n-2k))/4}.

A099488 Expansion of (1-x)^2/((1+x^2)(1-4x+x^2)).

Original entry on oeis.org

1, 2, 7, 28, 105, 390, 1455, 5432, 20273, 75658, 282359, 1053780, 3932761, 14677262, 54776287, 204427888, 762935265, 2847313170, 10626317415, 39657956492, 148005508553, 552364077718, 2061450802319, 7693439131560, 28712305723921
Offset: 0

Views

Author

Paul Barry, Oct 18 2004

Keywords

Comments

A Chebyshev transform of the sequence A081294 which has with g.f. (1-2x)/(1-4x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).

Links

Crossrefs

Cf. A099486, A099487.

Programs

  • Mathematica
    CoefficientList[Series[(1-x)^2/((1+x^2)(1-4x+x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{4,-2,4,-1},{1,2,7,28},30] (* Harvey P. Dale, Jun 23 2015 *)

Formula

a(n)=4a(n-1)-2a(n-2)+4a(n-3); a(n)=sum{k=0..n, (0^k-2sin(pi*k/2))((2+sqrt(3))^(n-k+1)-(2-sqrt(3))^(n-k+1))/(2*sqrt(3))}; a(n)=sum{k=0..n, (0^k-2sin(pi*k/2))A001353(n-k)}; a(n)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*(4^(n-2k)+0^(n-2k))/2}.

A290908 p-INVERT of the positive integers, where p(S) = 1 - 4*S^2.

Original entry on oeis.org

0, 4, 16, 56, 208, 780, 2912, 10864, 40544, 151316, 564720, 2107560, 7865520, 29354524, 109552576, 408855776, 1525870528, 5694626340, 21252634832, 79315912984, 296011017104, 1104728155436, 4122901604640, 15386878263120, 57424611447840, 214311567528244
Offset: 0

Views

Author

Clark Kimberling, Aug 17 2017

Keywords

Comments

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).
See A290890 for a guide to related sequences.

Links

Crossrefs

Cf. A000027, A099486, A290890.

Programs

  • Mathematica
    z = 60; s = x/(1 - x)^2; p = 1 - 4 s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)
u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A290908 *)
u/4 (* A099486 *)

Formula

G.f.: (4 x)/(1 - 4 x + 2 x^2 - 4 x^3 + x^4).
a(n) = 4*a(n-1) - 2*a(n-2) + 4*a(n-3) - a(n-4).
a(n) = 4*A099486(n) for n >= 0.

A099489 Expansion of (1-x+x^2)/((1+x^2)(1-4x+x^2)).

Original entry on oeis.org

1, 3, 11, 42, 157, 585, 2183, 8148, 30409, 113487, 423539, 1580670, 5899141, 22015893, 82164431, 306641832, 1144402897, 4270969755, 15939476123, 59486934738, 222008262829, 828546116577, 3092176203479, 11540158697340
Offset: 0

Views

Author

Paul Barry, Oct 18 2004

Keywords

Comments

A Chebyshev transform of the sequence A002001 which has with g.f. (1-x)/(1-4x). The image of G(x) under the Chebyshev transform is (1/(1+x^2))G(x/(1+x^2)).

Links

Crossrefs

Cf. A099486, A099487, A099488.

Programs

  • Mathematica
    CoefficientList[Series[(1-x+x^2)/((1+x^2)(1-4x+x^2)),{x,0,30}],x] (* or *_)
LinearRecurrence[{4,-2,4,-1},{1,3,11,42},30] (* Harvey P. Dale, Dec 28 2019 *)

Formula

a(n) = 4*a(n-1)-2*a(n-2)+4*a(n-3)-a(n-4). - corrected by Matthew House, Oct 22 2016
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*(3*4^(n-2*k)+0^(n-2*k))/4.
a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*((2+sqrt(3))^(n-k+1)-(2-sqrt(3))^(n-k+1))/(2*sqrt(3)).
a(n) = Sum_{k=0..n} (0^k-sin(Pi*k/2))*A001353(n-k+1).
a(n) = 3*A001353(n+1)/4 +A056594(n)/4. - R. J. Mathar, Sep 21 2012
Showing 1-5 of 5 results.

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