A127850 -id:A127850 - cp's OEIS frontend

A135950 Matrix inverse of triangle A022166.

1, -1, 1, 2, -3, 1, -8, 14, -7, 1, 64, -120, 70, -15, 1, -1024, 1984, -1240, 310, -31, 1, 32768, -64512, 41664, -11160, 1302, -63, 1, -2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1, 268435456, -534773760, 353730560, -99486720, 12850368, -777240, 21590, -255, 1

Offset: 0

Paul D. Hanna, Dec 08 2007

A022166 is the triangle of Gaussian binomial coefficients [n,k] for q = 2.
The coefficient [x^k] of Product_{i=1..n} (x-2^(i-1)). - Roger L. Bagula, Mar 20 2009
Triangle T(n,k), 0 <= k <= n, read by rows given by (-1, 1-q, -q^2, q-q^3, -q^4, q^2-q^5, -q^6, q^3-q^7, -q^8, ...) DELTA (1, 0, q, 0, q^2, 0, q^3, 0, q^4, 0, ...) (for q = 2) = (-1, -1, -4, -6, -16, -28, -64, -120, -256, ...) DELTA (1, 0, 2, 0, 4, 0, 8, 0, 16, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2013
Reversed rows of triangle A158474. - Werner Schulte, Apr 06 2019
T(n,k) = Sum mu(0,U) where the sum is taken over the subspaces U of GF(2)^n having dimension n-k and mu is the Moebius function of the poset of all subspaces of GF(2)^n. - Geoffrey Critzer, Jun 02 2024

			Triangle begins:
         1;
        -1,       1;
         2,      -3,        1;
        -8,      14,       -7,      1;
        64,    -120,       70,    -15,      1;
     -1024,    1984,    -1240,    310,    -31,    1;
     32768,  -64512,    41664, -11160,   1302,  -63,    1;
  -2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1; ...

G. C. Greubel, Rows n = 0..75 of triangle, flattened

Cf. A022166, A006125, A028361, A127850, A135951 (central terms), A158474.

max = 9; M = Table[QBinomial[n, k, 2], {n, 0, max}, {k, 0, max}] // Inverse; Table[M[[n, k]], {n, 1, max+1}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 08 2016 *)
p[x_, n_, q_] := (-1)^n*q^Binomial[n, 2]*QPochhammer[x, 1/q, n];
Table[CoefficientList[Series[p[x, n, 2], {x, 0, n}], x], {n, 0, 10}]// Flatten (* G. C. Greubel, Apr 15 2019 *)

T(n,k)=local(q=2,A=matrix(n+1,n+1,n,k,if(n>=k,if(n==1 || k==1, 1, prod(j=n-k+1, n-1, 1-q^j)/prod(j=1, k-1, 1-q^j))))^-1);A[n+1,k+1]

Unsigned column 0 equals A006125(n) = 2^(n*(n-1)/2).
Unsigned column 1 equals A127850(n) = (2^n-1)*2^(n*(n-1)/2)/2^(n-1).
Row sums equal 0^n.
Unsigned row sums equal A028361(n) = Product_{k=0..n} (1+2^k).
T(n,k) = (-1)^(n-k) * A022166(n,k) * 2^binomial(n-k,2) for 0 <= k <= n. - Werner Schulte, Apr 06 2019 [corrected by Werner Schulte, Dec 27 2021]
Sum_{n>=0} Sum_{k=0..n} T(n,k)y^k*x^n/A005329(n) = e(y*x)/e(x) where e(x) = Sum_{n>=0} x^n/A005329(n). - Geoffrey Critzer, Jun 02 2024

A225887 a(n) = A212205(2*n + 1).

Original entry on oeis.org

1, 4, 18, 86, 426, 2162, 11166, 58438, 309042, 1648154, 8851206, 47813790, 259585002, 1415431266, 7747200558, 42545600310, 234346445154, 1294260644906, 7165245015510, 39754745775886, 221009855334426, 1230909476804594, 6867024985408638, 38369226561522086

Offset: 0

Michael Somos, May 19 2013

From Peter Bala, Apr 23 2017: (Start)
a(n) is also the number of Schröder paths of semilength n (paths from (0, 0) to (2*n, 0), using only single steps northeast or southeast (steps (1, 1) or (1, -1)) or double steps east (steps (2, 0)), that never fall below the x-axis) in which the (2,0)-steps that are on the horizontal axis come in 3 colors (see Oste and Van der Jeugt, Section 7).
Example: a(2) = 18 because from the origin to the point (4,0) we have 3^2 = 9 paths of type HH, 3 paths of type HUD, 3 paths of type UDH as well as the paths UDUD, UUDD, and UHD.
It follows that the sequence may be calculated as the leading diagonal of the lower triangular array (T(n,k))n,k>=0 defined by the relations: T(n,0) = 1, T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k-1) for 1 <= k <= n-1 and T(n,n) = 3*T(n-1,n-1) + T(n,n-1). The array begins: [1], [1, 4], [1, 6, 18], [1, 8, 32, 86], [1, 10, 50, 168, 426].  (End)

			1 + 4*x + 18*x^2 + 86*x^3 + 426*x^4 + 2162*x^5 + 11166*x^6 + 58438*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.
Taras Goy and Mark Shattuck, Hessenberg-Toeplitz Matrix Determinants with Schröder and Fine Number Entries, Carpathian Math. Publ., Vol. 15 (2023), No. 2, 420-436. See Theorems 1 and 2.
R. Oste and J. Van der Jeugt, Motzkin paths, Motzkin polynomials and recurrence relations, Electronic Journal of Combinatorics 22(2) (2015), #P2.8. Section 7.

Cf. A001003, A006125, A127850, A131090, A151090, A212205, A111966.

a[ n_] := SeriesCoefficient[ 2 / (1 - 5 x + Sqrt[1 - 6 x + x^2]), {x, 0, n}]

a(n):=sum((k+1)*sum(binomial(j,n-k-j)*3^(-n+k+2*j)*2^(n-k-j)*binomial(n+1,j),j,0,n+1),k,0,n)/(n+1); /* Vladimir Kruchinin, Mar 13 2016 */

{a(n) = if( n<0, 0, polcoeff( 2 / (1 -  5*x + sqrt(1 - 6*x + x^2 + x * O(x^n))), n))}

G.f.: (-1 + 5*x + sqrt(1 - 6*x + x^2)) / (2 * (x - 6*x^2)) = 2 / (1 - 5*x + sqrt(1 - 6*x + x^2)).
G.f.: A(x) = 1 / (1 - 5*x + (x - 6*x^2) * A(x)) = 1 + x * A(x) * (5 - A(x) * (1 - 6*x)).
INVERT transform of A001003(n+1). INVERT transform is A134425.
HANKEL transform is A006125. HANKEL transform with 1 prepended is A127850(n+1).
BINOMIAL transform of A151090.
Conjecture: (n+1)*a(n) +3*(-4*n-1)*a(n-1) +(37*n-20)*a(n-2) +6*(-n+2)*a(n-3)=0. - R. J. Mathar, May 23 2014
a(n) = Sum_{k=0..n}((k+1)*Sum_{j=0..n+1}(binomial(j,n-k-j)*3^(-n+k+2*j)*2^(n-k-j)*binomial(n+1,j)))/(n+1). - Vladimir Kruchinin, Mar 13 2016
a(n) ~ (1+sqrt(2))^(2*n+5) / (2^(3/4)*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 13 2016
G.f.: 1/(1-3*x -x/(1-x -x/(1-x -x/(1-x - ... )))) (continued fraction) = 1/(1 - 3*x - x*S(x)), where S(x) is the generating function of the large Schröder numbers A001003. - Peter Bala, Apr 23 2017

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A135950 Matrix inverse of triangle A022166.

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A225887 a(n) = A212205(2*n + 1).

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A127851 a(n) has n 1's followed by C(n-1,2) 0's.

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