A162170 -id:A162170 - cp's OEIS frontend

A162171 Third column of A162170.

1, 3, 6, 20, 75, 336, 1708, 9792, 62325, 436480, 3334386, 27595776, 245951615, 2348666880, 23923317720, 258910994432, 2966901358185, 35886973648896, 456927138333790, 6108665873694720, 85555744482868275, 1252729007440396288, 19140289332506060676

Offset: 1

Mats Granvik, Jun 27 2009

J. A. Palacios, A. Bhaskar, F. Disanto and N. A. Rosenberg, Enumeration of binary trees compatible with a perfect phylogeny, J. Math. Biol. 84 (2022), 54.

Cf. A162170, A000111.

T(n, k) = if (k % 2, binomial(n-1, k-1) * (-1)^floor((n+k-1)/2), if (n==k, 1 , 0));
lista(nn) = {m = matrix(nn, nn, n, k, if (n>=k, T(n,k), 0)); m = m^(-1); for (n=3, nn, print1(m[n,3], ", "));} \\ Michel Marcus, Jun 17 2015

lista(nn) = { a = [1]; for(n = 2, nn, a = concat(a, sum(k = 1, j = floor(n/2), (-1)^(j+k) * binomial(n+1, 2*k) * a[2*k-1]))); print(a) } \\ Mike Tryczak, Jun 18 2015

a(n) = Sum_{k=1..floor(n/2)} (-1)^(floor(n/2)+k) * binomial(n+1, 2*k) * a(2*k-1) for n > 1. - Mike Tryczak, Jun 18 2015
a(n) = n*(n+1)/2 * A000111(n-1) (conjectured). - Mike Tryczak, Jun 17 2015
The above conjecture by Tryczak is correct. With an offset of 2, the e.g.f. is x^2/2!*(sec(x) + tan(x)). - Peter Bala, Sep 08 2021
a(n) is the number of ranked unlabeled binary tree shapes compatible with the binary perfect phylogeny (n,3). - Noah A Rosenberg, Jun 03 2022

Sequence corrected and extended by Mike Tryczak, Jun 17 2015

A062272 Boustrophedon transform of (n+1) mod 2.

Original entry on oeis.org

1, 1, 2, 5, 12, 41, 152, 685, 3472, 19921, 126752, 887765, 6781632, 56126201, 500231552, 4776869245, 48656756992, 526589630881, 6034272215552, 72989204937125, 929327412759552, 12424192360405961, 174008703107274752

Offset: 0

Frank Ellermann, Jun 16 2001

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Peter Luschny, An old operation on sequences: the Seidel transform.
J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory Ser. A, 76(1) (1996), 44-54 (Abstract, pdf, ps).
J. Millar, N. J. A. Sloane, and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory Ser. A, 76(1) (1996), 44-54.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

A000734 (binomial transform), a(2n+1)= A003719(n), a(2n)= A000795(n),
Cf. A062161 (n mod 2).
Row sums of A162170 minus A000035. - Mats Granvik, Jun 27 2009
Cf. A059841.

a062272 n = sum $ zipWith (*) (a109449_row n) $ cycle [1,0]
-- Reinhard Zumkeller, Nov 03 2013

s[n_] = Mod[n+1, 2]; t[n_, 0] := s[n]; t[n_, k_] := t[n, k] = t[n, k-1] + t[n-1, n-k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)

from itertools import accumulate, islice
def A062272_gen(): # generator of terms
    blist, m = tuple(), 0
    while True:
        yield (blist := tuple(accumulate(reversed(blist),initial=(m := 1-m))))[-1]
A062272_list = list(islice(A062272_gen(),40)) # Chai Wah Wu, Jun 12 2022

# Generalized algorithm of L. Seidel (1877)
def A062272_list(n) :
    R = []; A = {-1:0, 0:0}
    k = 0; e = 1
    for i in range(n) :
        Am = 1 if e == 1 else 0
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        R.append(A[e*i//2])
    return R
A062272_list(10) # Peter Luschny, Jun 02 2012

E.g.f.: (sec(x)+tan(x))cosh(x); a(n)=(A000667(n)+A062162(n))/2. - Paul Barry, Jan 21 2005
a(n) = Sum{k, k>=0} binomial(n, 2k)*A000111(n-2k). - Philippe Deléham, Aug 28 2005
a(n) = sum(A109449(n,k) * (1 - n mod 2): k=0..n). - Reinhard Zumkeller, Nov 03 2013

A162169 Exponential series expansion of (cos(x) - sin(x))cosh(tx) + sinh(t*x).

Original entry on oeis.org

1, -1, 1, -1, 0, 1, 1, 0, -3, 1, 1, 0, -6, 0, 1, -1, 0, 10, 0, -5, 1, -1, 0, 15, 0, -15, 0, 1, 1, 0, -21, 0, 35, 0, -7, 1, 1, 0, -28, 0, 70, 0, -28, 0, 1, -1, 0, 36, 0, -126, 0, 84, 0, -9, 1, -1, 0, 45, 0, -210, 0, 210, 0, -45, 0, 1, 1, 0, -55, 0, 330, 0, -462, 0, 165, 0, -11, 1

Offset: 1

Mats Granvik, Jun 27 2009

Previous name was: Signed version of Pascal's triangle.
Related to A000111 via its matrix inverse A162170.
For odd columns k, T(n, k) = binomial(n-1, k-1) * (-1)^floor((n+k-1)/2). For even columns, T(n, k) = 1 if n = k, otherwise 0. - Mike Tryczak, Jun 17 2015
From Peter Bala, Sep 08 2021: (Start)
In the notation of the Bala link, this is the array [[ cos(x) - sin(x), 1 ]] with inverse array A162170 = [[ sec(x) + tan(x), 1 ]].
In general, arrays of the form [[ G(x), 1 ]], where G(x) =  1 + g(1)*x + g(2)*x^2/2! + g(3)*x^3/3! + ... is an e.g.f., form a group with group law [[ G(x), 1 ]]*[[ F(x), 1 ]] = [[ G(x)*F_e(x) + F_o(x), 1 ]] and inverse array [[ G(x), 1 ]]^(-1) = [[ (1 - G_o(x))/G_e(x), 1 ]], where G_e(x) = (G(x) + G(-x))/2 and G_o(x) = (G(x) - G(-x))/2 are the even and odd parts of G(x). (End)

			Table begins:
   1;
  -1,    1;
  -1,    0,    1;
   1,    0,   -3,    1;
   1,    0,   -6,    0,    1;
  -1,    0,   10,    0,   -5,    1;
  -1,    0,   15,    0,  -15,    0,    1;
   1,    0,  -21,    0,   35,    0,   -7,    1;
   1,    0,  -28,    0,   70,    0,  -28,    0,    1;
  -1,    0,   36,    0, -126,    0,   84,    0,   -9,    1;
  -1,    0,   45,    0, -210,    0,  210,    0,  -45,    0,    1;
   1,    0,  -55,    0,  330,    0, -462,    0,  165,    0,  -11,    1;
.
As a symmetric triangle:
                                   1;
                               -1,    1;
                            -1,    0,    1;
                          1,    0,   -3,    1;
                       1,    0,   -6,    0,    1;
                   -1,    0,   10,    0,   -5,    1;
                -1,    0,   15,    0,  -15,    0,    1;
              1,    0,  -21,    0,   35,    0,   -7,    1;
           1,    0,  -28,    0,   70,    0,  -28,    0,    1;
       -1,    0,   36,    0, -126,    0,   84,    0,   -9,    1;
    -1,    0,   45,    0, -210,    0,  210,    0,  -45,    0,    1;
  1,    0,  -55,    0,  330,    0, -462,    0,  165,    0,  -11,    1;

Peter Bala, Matrices with repeated columns - the generalised Appell groups

Cf. A007318, A162170.

=if(or(mod(row()-column();4)=1;mod(row()-column();4)=2);-1;1)*if(row()>=column();combin(row()-1;column()-1);0)*if(and(row()>column();mod(column();2)=0);0;1)

egf := (cos(x) - sin(x))*cosh(t*x) + sinh(t*x):
ser := n -> series(egf, x, n+1): c := n -> n!*coeff(ser(n), x, n):
A162169row := n -> seq(coeff(c(n), t, k), k=0..n):
for n from 0 to 9 do A162169row(n) od; # Peter Luschny, Sep 18 2021

nn=12; Flatten[Table[Table[If[Or[Mod[n - k, 4] == 1, Mod[n - k, 4] == 2], -1, 1]*If[n >= k, Binomial[n - 1, k - 1], 0]*If[And[n > k, Mod[k, 2] == 0], 0, 1], {k, 1, n}], {n, 1, nn}]] (* Mats Granvik, Nov 25 2017 *)

T(n, k) = if (k % 2, binomial(n-1, k-1) * (-1)^floor((n+k-1)/2), if (n==k, 1 , 0));
tabl(nn) = {for (n=1, nn, for (k=1, n, print1(T(n,k), ", ");); print(););} \\ Michel Marcus, Jun 17 2015

E.g.f.: (cos(x) - sin(x))*cosh(t*x) + sinh(t*x) = 1 + (-1 + t)*x + (-1 + t^2)*x^2/2! + (1 - 3^t^2 + t^3)*x^3/3! + .... - Peter Bala, Sep 08 2021

New name using a formula of Peter Bala from Peter Luschny, Sep 18 2021

A173253 Partial sums of A000111.

Original entry on oeis.org

1, 2, 3, 5, 10, 26, 87, 359, 1744, 9680, 60201, 413993, 3116758, 25485014, 224845995, 2128603307, 21520115452, 231385458428, 2636265133869, 31725150246701, 402096338484226, 5353594391608322, 74702468784746223, 1090126355291598575, 16604660518848685480

Offset: 0

Jonathan Vos Post, Feb 14 2010

nonn

Partial sums of Euler or up/down numbers. Partial sums of expansion of sec x + tan x. Partial sums of number of alternating permutations on n letters.

			a(22) = 1 + 1 + 1 + 2 + 5 + 16 + 61 + 272 + 1385 + 7936 + 50521 + 353792 + 2702765 + 22368256 + 199360981 + 1903757312 + 19391512145 + 209865342976 + 2404879675441 + 29088885112832 + 370371188237525 + 4951498053124096 + 69348874393137901.

Alois P. Heinz, Table of n, a(n) for n = 0..485

Cf. A000111, A000364, A000182, A008280, A008281, A008282, A010094, A059720, A008970, A109449, A162170.

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= proc(n) option remember;
      `if`(n<0, 0, a(n-1))+ b(n, 0)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 27 2017

With[{nn=30},Accumulate[CoefficientList[Series[Sec[x]+Tan[x],{x,0,nn}],x] Range[0,nn]!]] (* Harvey P. Dale, Feb 26 2012 *)

from itertools import accumulate
def A173253(n):
    if n<=1:
        return n+1
    c, blist = 2, (0,1)
    for _ in range(n-1):
        c += (blist := tuple(accumulate(reversed(blist),initial=0)))[-1]
    return c # Chai Wah Wu, Apr 16 2023

a(n) = SUM[i=0..n] A000111(i) = SUM[i=0..n] (2^i|E(i,1/2)+E(i,1)| where E(n,x) are the Euler polynomials).
G.f.: (1 + x/Q(0))/(1-x),m=+4,u=x/2, where Q(k) = 1 - 2*u*(2*k+1) - m*u^2*(k+1)*(2*k+1)/( 1 - 2*u*(2*k+2) - m*u^2*(k+1)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013
G.f.: 1/(1-x) + T(0)*x/(1-x)^2, where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(1-x*(k+1))*(1-x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2013
a(n) ~ 2^(n+2)*n!/Pi^(n+1). - Vaclav Kotesovec, Oct 27 2016

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A162171 Third column of A162170.

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A062272 Boustrophedon transform of (n+1) mod 2.

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A162169 Exponential series expansion of (cos(x) - sin(x))*cosh(t*x) + sinh(t*x).

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A173253 Partial sums of A000111.

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A162169 Exponential series expansion of (cos(x) - sin(x))cosh(tx) + sinh(t*x).