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.

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A090158 Odd-indexed terms of the binomial transform equals 1 and the even-indexed terms of the second binomial transform equals 1.

Original entry on oeis.org

1, 0, -3, 9, -15, 15, -63, 399, -255, -7425, -1023, 355839, -4095, -22360065, -16383, 1903790079, -65535, -209865211905, -262143, 29088885637119, -1048575, -4951498051026945, -4194303, 1015423886515240959, -16777215, -246921480190174429185
Offset: 0

Views

Author

Paul D. Hanna, Nov 22 2003

Keywords

Comments

Compare the first and 2nd binomial transforms of this sequence:
first binomial={1,1,-2,1,4,1,-62,1,1384,1,-50522,1,2702764,..};
2nd binomial={1,2,1,-1,1,17,1,-271,1,7937,1,-353791,..};
to that of the first and 2nd binomial transforms of A090145:
first binomial of A090145={1,0,1,-3,1,15,1,-273,1,7935,1,..};
2nd binomial of A090145={1,1,2,1,-4,1,62,1,-1384,1,50522,..}.
Comparison reveals this e.g.f. relation of the two sequences:
e.g.f.: exp(x)*G090158(x) + exp(2x)*G090145(x) = 2 + 2*sinh(x);
e.g.f.: exp(2*x)*G090158(x) - exp(x)*G090145(x) = 2*sinh(x);
thus G090158(x) = 2*(1+sinh(x) + exp(x)*sinh(x))/(exp(x)*(1+exp(2*x)))
G090145(x) = 2*((1+sinh(x))*exp(x) - sinh(x))/(exp(x)*(1+exp(2*x))).

Crossrefs

Cf. A090145.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[2 (1+Sinh[x]+Exp[x]Sinh[x])/ (Exp[x] (1+ Exp[2x])),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 13 2016 *)

Formula

E.g.f.: 2*(1 + sinh(x) + exp(x)*sinh(x)) / (exp(x)*(1 + exp(2*x))).
a(2n) = 1 - 2^(2n);
1 = sum_{k=0..2n-1} C(2n-1, k)*a(k);
1 = sum_{k=0..2n} 2^(2n-k)*C(2n, k)*a(k).

A217714 Modified Euler numbers.

Original entry on oeis.org

1, 0, -2, -3, 4, 15, -62, -273, 1384, 7935, -50522, -353793, 2702764, 22368255, -199360982, -1903757313, 19391512144, 209865342975, -2404879675442, -29088885112833, 370371188237524, 4951498053124095, -69348874393137902, -1015423886506852353, 15514534163557086904, 246921480190207983615, -4087072509293123892362
Offset: 0

Views

Author

Paul Curtz, Mar 21 2013

Keywords

Comments

a(n) and differences are:
1, 0, -2, -3, 4, 15, -62;
-1, -2, -1, 7, 11, -77;
-1, 1, 8, 4, -88;
2, 7, -4, -92;
5, -11, -88;
-16, -77;
-61;
(See the array in A163982(n) and the comments/examples in A090158 and A090145.)
The absolute values of the first column are A000111(n).
The first column can be found via the Akiyama-Tanigawa algorithm. See the chapter on the Seidel triangle in Wikipedia's Bernoulli Number.

Examples

			a(0) =   1;
a(1) =   1 -   1 = 0;
a(2) =  -1 -   2 +   1 = -2;
a(3) =   2 -   3 -   3 +   1 = -3;
a(4) =   5 +   8 -   6 -   4 +  1 = 4;
a(5) = -16 +  25 +  20 -  10 -  5 +  1 = 15;
a(6) = -61 -  96 +  75 +  40 - 15 -  6 + 1 = -62;
a(7) = 272 - 427 - 336 + 175 + 70 - 21 - 7 + 1 = -273; - _Philippe Deléham_, Oct 27 2013
G.f. = 1 - 2*x^2 - 3*x^3 + 4*x^4 + 15*x^5 - 62*x^6 - 273*x^7 + ...

Links

Crossrefs

Cf. A000111, A163982.

Programs

  • Mathematica
    a[n_] := 2^n* EulerE[n, 1] + EulerE[n] - 1; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Mar 21 2013 *)

Formula

a(n) = -A163982(n) - 1.
a(n) = Sum_{k=0..n} A109449(n,k)*floor((n-k+1)/2). - Philippe Deléham, Oct 27 2013
E.g.f.: 1/cosh(x) + tanh(x) + 1 - exp(x). - Sergei N. Gladkovskii, Nov 10 2014

Extensions

More terms from Jean-François Alcover, Mar 21 2013

A090336 Odd-indexed terms of the first binomial transform equals 1 and the even-indexed terms of the third binomial transform equals 1, with a(0)=1.

Original entry on oeis.org

1, 0, -8, 24, 64, -480, -3968, 34944, 354304, -4062720, -51734528, 724568064, 11070521344, -183240744960, -3266330329088, 62382319632384, 1270842139869184, -27507470234419200, -630424777639067648, 15250953398036987904, 388362339077349965824
Offset: 0

Views

Author

Paul D. Hanna, Nov 25 2003

Keywords

Examples

			Successive binomial transforms are:
0th: {1,0,-8,24,64,-480,-3968,34944,354304,-4062720,...}
1st: {1,1,-7,1,113,1,-5527,1,501473,1,-73163047,1,...}
2nd: {1,2,-4,-16,80,512,-3904,-34816,354560,4063232,...}
3rd: {1,3,1,-21,1,723,1,-49221,1,5746083, 1,...} and
4th: {1,4,8,-8,-64,544,3968, -34688,-354304,4063744,...}
The sum of this sequence with its 4th binomial transform equals {2,4,0,16,0,64,0,64,0,256,0,1024,...}, which has e.g.f.: 2+2sinh(2x).
This describes the e.g.f.: A+exp(4x)*A=2+2sinh(2x).

Crossrefs

Cf. A090145, A090158.

Programs

  • Mathematica
    CoefficientList[Series[2*(1+Sinh[2*x])/(1+E^(4*x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 06 2014 *)
Table[2^(n - 1)*(EulerE[n]-2^n (EulerE[n, -1/2] - 2 EulerE[n, 0])), {n, 0, 20}] (* Benedict W. J. Irwin, May 26 2016 *)

Formula

E.g.f.: 2*(1+sinh(2*x))/(1+exp(4*x)).
a(n) ~ n! * (cos(Pi*n/2)-sin(Pi*n/2)) / (Pi/4)^(n+1). - Vaclav Kotesovec, Mar 06 2014
a(n) = 2^(n-1)*(EulerE(n) - 2^n*(EulerE(n,-1/2) - 2*EulerE(n,0))). - Benedict W. J. Irwin, May 26 2016
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