A034428 -id:A034428 - cp's OEIS frontend

A131281 Expansion of e.g.f.: 2(x-1)tan(x/2+Pi/4)-x^2+2.

0, 0, 0, 2, 6, 18, 70, 310, 1582, 9058, 57678, 403878, 3085478, 25535378, 227589206, 2173314806, 22137209694, 239580726978, 2745392996254, 33207657441094, 422813028038230, 5652593799727858, 79168165551184422, 1159200449070638742, 17711278225214739086

Offset: 0

N. J. A. Sloane, Oct 30 2007

nonn

Y. Sano, The principal numbers of K. Saito for the types A_l, D_l and E_l, Discr. Math., 307 (2007), 2636-2642.

Essentially the same as 2*A034428.

With[{nn=30},CoefficientList[Series[2(x-1)Tan[x/2+Pi/4]-x^2+2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 11 2023 *)

E.g.f. E(x)=2*(x-1)*tan(x/2+Pi/4)-x^2+2 = 2*x - x^2 + 4*x*(x-1)/(Q(0)-x) where Q(k) = 4*k + 2 - x^2/Q(k+1); (continued fraction, 1-step).- Sergei N. Gladkovskii, Jun 22 2012

a(n) ~ n! * 2^(n + 2) * (Pi - 2) / Pi^(n + 1). - Vaclav Kotesovec, Mar 12 2019

Definition clarified by Harvey P. Dale, Jun 11 2023

A131656 Principal number of K. Saito for tree of type E_n.

Original entry on oeis.org

-1, 3, 0, 3, 5, 18, 66, 298, 1511, 8670, 55168, 386394, 2951673, 24428654, 217723390, 2079109386, 21177620171, 229195610430, 2626388037372, 31768201320634, 404485298533085, 5407570127090958, 75736453324821754, 1108952444876609898, 16943545270848408495

Offset: 0

N. J. A. Sloane, Oct 30 2007

sign

Y. Sano, The principal numbers of K. Saito for the types A_l, D_l and E_l, Discr. Math., 307 (2007), 2636-2642.

Same as A131611 except for leading terms. Cf. A000111, A131281, A034428.

A172170 1 followed by the duplicated entries of A090368.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 5, 7, 7, 3, 3, 11, 11, 13, 13, 3, 3, 17, 17, 19, 19, 3, 3, 23, 23, 5, 5, 3, 3, 29, 29, 31, 31, 3, 3, 5, 5, 37, 37, 3, 3, 41, 41, 43, 43, 3, 3, 47, 47, 7, 7, 3, 3, 53, 53, 5, 5, 3, 3, 59, 59, 61, 61, 3, 3, 5, 5, 67, 67, 3, 3, 71, 71, 73, 73, 3, 3, 7, 7, 79, 79, 3, 3, 83, 83, 5, 5

Offset: 0

Paul Curtz, Jan 28 2010

nonn

We start from the expansion tan(x)+sec(x) = sum_{n>=1} A099612(n)/A099617(n) * x^n with Taylor coefficients 1, 1, 1/2, 1/3, 5/24, 2/15,...
The first differences of this sequence of fractions are 0, -1/2, -1/6, -1/8, -3/40, -7/144, -31/1008, -113/5760,... which is 0 followed by the negated ratios A034428(n)/(n+1)! = 0, -1/2, -1/6, -3/24, -9/120,....
(The factorial follows because A034428 is obtained by multiplying with 1-x to generate first differences of the o.g.f. and then moving on to the e.g.f.)
The common multiple to reduce numerator and denominator of A034428(n)/A000142(n+1) to the standard coprime representation is this sequence here.

a(2n+1)=a(2n+2) = A090368(n), n>=0.

A211182 E.g.f. 1-x(1-x)(tan(x)+sec(x)).

Original entry on oeis.org

1, -1, 0, 3, 4, 15, 54, 245, 1240, 7119, 45290, 317229, 2423268, 20055607, 178747646, 1706919045, 17386518448, 188166282399, 2156226542802, 26081233464413, 332076574410940, 4439536794401415, 62178531797006438, 910433903838620853, 13910405388847664904

Offset: 0

LtC. R. Scott Patterson (aa737drvr(AT)comcast.net) and Robert G. Wilson v, Feb 02 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..202

Cf. A034428.

With[{nn = 24}, CoefficientList[ Series[1 - x (1 - x) (Tan[x] + Sec[x]), {x, 0, nn}], x] Range[0, nn]!]
Join[{1, -1, 0}, Table[-n * (((n-1)*EulerE[n-2] + 2^(n-1)*EulerE[n-1, 0]) * Cos[Pi*n/2] + (EulerE[n-1] - 2^(n-2)*(n-1)*EulerE[n-2, 0]) * Sin[Pi*n/2]), {n, 3, 30}]] (* Benedict W. J. Irwin, Jul 28 2016 *)

a(n) ~ n! * (Pi-2)*2^n/Pi^n. - Vaclav Kotesovec, Feb 12 2013

A337443 E.g.f.: (1 + x) * exp(x) / (sec(x) + tan(x)).

Original entry on oeis.org

1, 1, 0, -1, -4, -5, -20, 9, -208, 855, -6180, 41549, -321792, 2651155, -23664420, 225865769, -2301032256, 24901626095, -285356879940, 3451591584869, -43947119045600, 587529302036875, -8228722825167940, 120487046847246049, -1840906518665985824

Offset: 0

Ilya Gutkovskiy, Aug 27 2020

sign

Inverse boustrophedon transform of positive natural numbers.

Index entries for sequences related to boustrophedon transform

Cf. A000027, A000111, A000737, A034428, A231179.

nmax = 24; CoefficientList[Series[(1 + x) Exp[x]/(Sec[x] + Tan[x]), {x, 0, nmax}], x] Range[0, nmax]!
t[n_, 0] := n + 1; t[n_, k_] := t[n, k] = t[n, k - 1] - t[n - 1, n - k]; a[n_] := t[n, n]; Table[a[n], {n, 0, 24}]

from itertools import count, islice, accumulate
from operator import sub
def A337443_gen(): # generator of terms
    blist = tuple()
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),func=sub,initial=i)))[-1]
A337443_list = list(islice(A337443_gen(),30)) # Chai Wah Wu, Jun 11 2022

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * (k+1) * A000111(n-k).

cp's OEIS Frontend

A131281 Expansion of e.g.f.: 2*(x-1)*tan(x/2+Pi/4)-x^2+2.

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A131656 Principal number of K. Saito for tree of type E_n.

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A172170 1 followed by the duplicated entries of A090368.

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A211182 E.g.f. 1-x*(1-x)*(tan(x)+sec(x)).

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A337443 E.g.f.: (1 + x) * exp(x) / (sec(x) + tan(x)).

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A131281 Expansion of e.g.f.: 2(x-1)tan(x/2+Pi/4)-x^2+2.

A211182 E.g.f. 1-x(1-x)(tan(x)+sec(x)).