A153881 -id:A153881 - cp's OEIS frontend

A378225 Dirichlet inverse of A067824.

1, -2, -2, 0, -2, 2, -2, 0, 0, 2, -2, 0, -2, 2, 2, 0, -2, 0, -2, 0, 2, 2, -2, 0, 0, 2, 0, 0, -2, -2, -2, 0, 2, 2, 2, 0, -2, 2, 2, 0, -2, -2, -2, 0, 0, 2, -2, 0, 0, 0, 2, 0, -2, 0, 2, 0, 2, 2, -2, 0, -2, 2, 0, 0, 2, -2, -2, 0, 2, -2, -2, 0, -2, 2, 0, 0, 2, -2, -2, 0, 0, 2, -2, 0, 2, 2, 2, 0, -2, 0, 2, 0, 2, 2, 2, 0, -2

Offset: 1

Antti Karttunen, Nov 25 2024

sign

Möbius transform of A153881.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A008683, A067824, A153881.

Cf. also A378224.

A074206(n) = if(n>1, sumdiv(n, i, if(iA074206(i))), n); \\ From A074206
A067824(n) = sumdiv(n,d,A074206(d));
memoA378225 = Map();
A378225(n) = if(1==n,1,my(v); if(mapisdefined(memoA378225,n,&v), v, v = -sumdiv(n,d,if(dA067824(n/d)*A378225(d),0)); mapput(memoA378225,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA067824(n/d) * a(d).
a(n) = Sum_{d|n} A008683(n/d)*A153881(d).
Dirichlet g.f.: (2 - zeta(s)) / zeta(s). [See Dec 30 2018 formula in A067824]

A134824 Generated by reverse of Schroeder II o.g.f.

Original entry on oeis.org

0, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 0

Wolfdieter Lang, Nov 13 2007

The o.g.f. S(x) for A001003 (Schroeder II) satisfies 2*S^2(x) + (1+x)*S(x) + x = 0.

Using the Lagrange series for y=S(x) with y=0+x*(y/A(y)) leads to the formula for Schroeder II numbers involving the Narayana triangle A001263. See the Narayana comment by B. Cloitre under A001003 and a multiple differentiation formula given there.

Index entries for linear recurrences with constant coefficients, signature (1).

If the initial 0 is omitted, we get A153881.

G.f.: x*(1-2*x)/(1-x).

a(0)=0,a(1)=1, a(n)=-1, n>=2.

A251635 Riordan array (1-2x,x), inverse of Riordan array (1/(1-2x), x) = A130321.

Original entry on oeis.org

1, -2, 1, 0, -2, 1, 0, 0, -2, 1, 0, 0, 0, -2, 1, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0

Offset: 0

Wolfdieter Lang, Jan 10 2015

This is a simple Riordan array, an infinite lower triangular matrix. It is the inverse matrix of A130321 (with zeros above the diagonal).
Row sums have o.g.f. (1-2*x)/(1-x) and give 1, repeat(-1), i.e., A153881(n+1), n >= 0.
Alternate row sums have o.g.f. (1-2*x)/(1+x) and give 1, repeat(-3,3), i.e., (-1)^n*A122553(n).

			The triangle T(n, k) begins:
n\k  0  1  2  3  4  5  6  7  8  9 10 ...
0:   1
1:  -2  1
2:   0 -2  1
3:   0  0 -2  1
4:   0  0  0 -2  1
5:   0  0  0  0 -2  1
6:   0  0  0  0  0 -2 1
7:   0  0  0  0  0  0 -2  1
8:   0  0  0  0  0  0  0 -2  1
9:   0  0  0  0  0  0  0  0 -2  1
10:  0  0  0  0  0  0  0  0  0 -2  1
...

Cf. A130321, A153881, A122553.

a251635 n k = a251635_tabl !! n !! k
a251635_row n = a251635_tabl !! n
a251635_tabl = [1] : iterate (0 :) [-2, 1]
-- Reinhard Zumkeller, Jan 11 2015

T(n, k) = 0 if n < k and k = 0..(n-2) for n >= 2, and T(n, n) = 1 and T(n, n-1) = -2.
G.f. for row polynomials P(n, x) = -2^x^(n-1) + x^n is (1-2*z)/(1-x*z).
G.f. for k-th column: (1-2*x)*x^k, k >= 0.

A154959 Triangle read by rows. Signed version of A008277.

Original entry on oeis.org

1, -1, 1, -1, -3, 1, -1, -7, -6, 1, -1, -15, -25, -10, 1, -1, -31, -90, -65, -15, 1, -1, -63, -301, -350, -140, -21, 1, -1, -127, -966, -1701, -1050, -266, -28, 1, -1, -255, -3025, -7770, -6951, -2646, -462, -36, 1

Offset: 1

Mats Granvik, Jan 18 2009

Main diagonal positive, the rest of the terms negative. Matrix inverse of this triangle is A154960. Signs in columns as in A153881.

A298673 Inverse matrix of A135494.

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 26, 19, 6, 1, 236, 170, 55, 10, 1, 2752, 1966, 645, 125, 15, 1, 39208, 27860, 9226, 1855, 245, 21, 1, 660032, 467244, 155764, 32081, 4480, 434, 28, 1, 12818912, 9049584, 3031876, 635124, 92001, 9576, 714, 36, 1, 282137824, 198754016, 66845340, 14180440, 2108085, 230097, 18690, 1110, 45, 1

Offset: 1

Tom Copeland, Jan 24 2018

Since this is the inverse matrix of A135494 with row polynomials q_n(t), first introduced in that entry by R. J. Mathar, and the row polynomials p_n(t) of this entry are a binomial Sheffer polynomial sequence, the row polynomials of the inverse pair are umbral compositional inverses, i.e., p_n(q.(t)) = q_n(p.(t)) = t^n. For example, p_3(q.(t)) = 4q_1(t) + 3q_2(t) + q_3(t) = 4t + 3(-t + t^2) + (-t -3t^2 +t^3) = t^3. In addition, both sequences possess the umbral convolution property (p.x) + p.(y))^n = p_n(x+y) with p_0(t) = 1.

This is the inverse of the Bell matrix generated by A153881; for the definition of the Bell matrix see the link. - Peter Luschny, Jan 26 2018

			Matrix begins as
     1;
     1;    1;
     4,    3,    1;
    26,   19,    6,    1;
   236,  170,   55,   10,    1;
  2752, 1966,  645,  125,   15,    1;

Peter Luschny, The Bell transform
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.

Cf. A000311, A000169, A000670, A006351, A135494.

# The function BellMatrix is defined in A264428. Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n=0, 1, -1), 9): MatrixInverse(%); # Peter Luschny, Jan 26 2018

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, If[n == 0, 1, -1]], rows = 12] // Inverse;
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

E.g.f.: e^[p.(t)x] = e^[t*h(x)] = exp[t*[(x-1)/2 + T{ (1/2) * exp[(x-1)/2] }], where T is the tree function of A000169 related to the Lambert function. h(x) = sum(j=1,...) A000311(j) * x^j / j! = exp[xp.'(0)], so the first column of this entry's matrix is A000311(n) for n > 0 and the second column of the full matrix for p_n(t) to n >= 0. The compositional inverse of h(x) is h^(-1)(x) = 1 + 2x - e^x.
The lowering operator is L = h^(-1)(D) = 1 + 2D - e^D with D = d/dt, i.e., L p_n(t) = n * p_(n-1)(t). For example, L p_3(t) = (D - D^2! - D^3/3! - ...) (4t + 6t^ + t^3) = 3 (t + t^2) = 3 p_2(t).
The raising operator is R = t * 1/[d[h^(-1)(D)]/dD] = t * 1/[2 - e^D)] = t (1 + D + 3D^2/2! + 13D^3/3! + ...). The coefficients of R are A000670. For example, R p_2(t) = t (1 + D + 3D^2/2!  + ...)  (t + t^2) = 4t + 3t^2 + t^3   = p_3(t).
The row sums are A006351, or essentially 2*A000311.
Conjectures from Mikhail Kurkov, Mar 01 2025: (Start)
T(n,k) = Sum_{j=0..n-k} binomial(n+j-1, k-1)*A269939(n-k, j) for 1 <= k <= n.
T(n,k) = A(n-1,k,0) for n > 0, k > 0 where A(n,k,q) = A(n-1,k,q+1) + 2*(q+1)!*Sum_{j=0..q} A(n-1,k,j)/j! for n >= 0, k > 0, q >= 0 with A(0,k,q) = Stirling1(q+1,k) for k > 0, q >= 0 (see A379458). In other words, T(n,k) = Sum_{j=0}^{n-1} A379460(n-j-1,j)*Stirling1(j+1,k) for n > 0, k > 0.
Recursion for the n-th row (independently of other rows): T(n,k) = 1/(n-k)*Sum_{j=2..n-k+1} b(j-1)*binomial(-k,j)*T(n,k+j-1)*(-1)^j for 1 <= k < n with T(n,n) = 1 where b(n) = 1 + 4*Sum_{i=1..n} A135148(i).
Recursion for the k-th column (independently of other columns): T(n,k) = 1/(n-k)*Sum_{j=2..n-k+1} c(j-1)*binomial(n,j)*T(n-j+1,k) for 1 <= k < n with T(n,n) = 1 where c(n) = A000311(n+1) + (n-1)*A000311(n). (End)

A373164 Triangle read by rows: the exponential almost-Riordan array ( 1 | 2 - exp(x), x ).

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, -1, -2, 1, 0, -1, -3, -3, 1, 0, -1, -4, -6, -4, 1, 0, -1, -5, -10, -10, -5, 1, 0, -1, -6, -15, -20, -15, -6, 1, 0, -1, -7, -21, -35, -35, -21, -7, 1, 0, -1, -8, -28, -56, -70, -56, -28, -8, 1, 0, -1, -9, -36, -84, -126, -126, -84, -36, -9, 1

Offset: 0

Stefano Spezia, May 26 2024

			The triangle begins:
  1;
  0,  1;
  0, -1,  1;
  0, -1, -2,   1;
  0, -1, -3,  -3,   1;
  0, -1, -4,  -6,  -4,   1;
  0, -1, -5, -10, -10,  -5,   1;
  0, -1, -6, -15, -20, -15,  -6,  1;
  0, -1, -7, -21, -35, -35, -21, -7, 1;
  ...

Y. Alp and E. G. Kocer, Exponential Almost-Riordan Arrays, Results Math 79, 173 (2024). See page 8.

Cf. A000012 (right diagonal), A024000 (subdiagonal), A122958 (row sums), A153881 (k=1).

Triangle A154926 with 1st column A000007.

T[n_,0]:=KroneckerDelta[n,0]; T[n_,k_]:=(n-1)!/(k-1)!SeriesCoefficient[(2-Exp[x])x^(k-1),{x,0,n-1}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten

T(n,0) = A000007(n); T(n,k) = (n-1)!/(k-1)! * [x^(n-1)] (2-exp(x))*x^(k-1).

cp's OEIS Frontend

A378225 Dirichlet inverse of A067824.

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A134824 Generated by reverse of Schroeder II o.g.f.

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A251635 Riordan array (1-2*x,x), inverse of Riordan array (1/(1-2*x), x) = A130321.

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A154959 Triangle read by rows. Signed version of A008277.

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A298673 Inverse matrix of A135494.

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A373164 Triangle read by rows: the exponential almost-Riordan array ( 1 | 2 - exp(x), x ).

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Mathematica

Formula

A251635 Riordan array (1-2x,x), inverse of Riordan array (1/(1-2x), x) = A130321.