A126930 -id:A126930 - cp's OEIS frontend

A124644 Triangle read by rows. T(n, k) = binomial(n, k) * CatalanNumber(n - k).

1, 1, 1, 2, 2, 1, 5, 6, 3, 1, 14, 20, 12, 4, 1, 42, 70, 50, 20, 5, 1, 132, 252, 210, 100, 30, 6, 1, 429, 924, 882, 490, 175, 42, 7, 1, 1430, 3432, 3696, 2352, 980, 280, 56, 8, 1, 4862, 12870, 15444, 11088, 5292, 1764, 420, 72, 9, 1, 16796, 48620, 64350, 51480, 27720

Offset: 0

Farkas Janos Smile (smile_farkasjanos(AT)yahoo.com.au), Dec 21 2006

Equal to A091867*A007318. - Philippe Deléham, Dec 12 2009
Exponential Riordan array [exp(2x)*(Bessel_I(0,2x)-Bessel_I(1,2x)),x]. - Paul Barry, Mar 03 2011
From Tom Copeland, Nov 04 2014: (Start)
O.g.f: G(x,t) = C[Pinv(x,t)] =  {1 - sqrt[1 - 4 *x /(1-x*t)]}/2 where C(x) = [1 - sqrt(1-4x)]/2, an o.g.f. for the shifted Catalan numbers A000108 with inverse Cinv(x) = x*(1-x), and Pinv(x,t)= -P(-x,t) = x/(1-t*x) with inverse P(x,t) = 1/(1+t*x). This puts this array in a family of arrays formed from the composition of C and P and their inverses. -G(-x,t) is the comp. inverse of the o.g.f. of A030528.
This is an Appell sequence with lowering operator d/dt p(n,t) = n*p(n-1,t) and (p(.,t)+a)^n = p(n,t+a). The e.g.f. has the form e^(x*t)/w(t) where 1/w(t) is the e.g.f. of the first column, which is the Catalan sequence A000108. (End)

			From _Paul Barry_, Jan 28 2009: (Start)
Triangle begins
   1,
   1,  1,
   2,  2,  1,
   5,  6,  3,  1,
  14, 20, 12,  4,  1,
  42, 70, 50, 20,  5,  1 (End)

Indranil Ghosh, Rows 0..125, flattened
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.

Cf. A098474 (mirror image), A000108, A091867, A030528, A104597.

Row sums give A007317(n+1).

m:=n->binomial(2*n, n)/(n+1): T:=proc(n, k) if k<=n then binomial(n, k)*m(n-k) else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od;

Table[Binomial[n, #] Binomial[2 #, #]/(# + 1) &[n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* or *)
Table[Abs[(-1)^k*CatalanNumber[#] Pochhammer[-n, #]/#!] &[n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Feb 17 2017 *)

def A124644(n,k):
    return (-1)^(n-k)*catalan_number(n-k)*rising_factorial(-n,n-k)/factorial(n-k)
for n in range(7): [A124644(n,k) for k in (0..n)] # Peter Luschny, Feb 05 2015

T(n,k) = [x^(n-k)]F(-n,n-k+1;1;-1-x). - Paul Barry, Sep 05 2008
G.f.: 1/(1-xy-x/(1-x/(1-xy-x/(1-x/(1-xy-x/(1-x.... (continued fraction). - Paul Barry, Jan 06 2009
G.f.: 1/(1-x-xy-x^2/(1-2x-xy-x^2/(1-2x-xy-x^2/(1-.... (continued fraction). - Paul Barry, Jan 28 2009
T(n,k) = Sum_{i = 0..n} C(n,i)*(-1)^(n-i)*Sum{j = 0..i} C(j,k)*C(i,j)*A000108(i-j). - Paul Barry, Aug 03 2009
Sum_{k = 0..n} T(n,k)*x^k = A126930(n), A005043(n), A000108(n), A007317(n+1), A064613(n), A104455(n) for x = -2, -1, 0, 1, 2, 3 respectively. T(n,k)= A007318(n,k)*A000108(n-k). - Philippe Deléham, Dec 12 2009
E.g.f.: exp(2*x + x*y)*(Bessel_I(0,2*x) - Bessel_I(1,2*x)). - Paul Barry, Mar 10 2010
From Tom Copeland, Nov 08 2014: (Start)
O.g.f.: G(x,t) = C[P(x,t)] = [1 - sqrt(1-4*x / (1-t*x))] / 2 = Sum_{n >= 1} (C. +  t)^(n-1) * x^n] = x + (1 + t) x^2 + (2 + 2t + t^2) x^3 + ... umbrally, where (C.)^n = C_n = (1,1,2,5,8,...) = A000108(x), C(x)= x*A000108(x)= G(x,0), and P(x,t) = x/(1 + t*x), a special linear fractional (Mobius) transformation. P(x,-t)= -P(-x,t) is the inverse of P(x,t).
Inverse o.g.f.: Ginv(x,t) = P[Cinv(x),-t] = x*(1-x) / [1 - t*x(1-x)] = -A030528(-x,t), where Cinv(x) = x*(1-x) is the inverse of C(x).
G(x,t) = x*A091867(x,t+1), and Ginv(x,t) = x*A104597(x,-(t+1)). (End)
T(n, k) = (-1)^(n-k)*Catalan(n-k)*Pochhammer(-n,n-k)/(n-k)!. - Peter Luschny, Feb 05 2015
Recurrence: T(n, 0) = Catalan(n) = 1/(n+1)*binomial(2*n, n) and, for 1 <= k <= n, T(n, k) = (n/k) * T(n-1, k-1). - Peter Bala, Feb 04 2024

Name brought in line with the Maple program by Peter Luschny, Jun 21 2023

A129174 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n such that the sum of the peak-abscissae is k (0 <= k <= n^2).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 2, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 3, 5, 5, 7, 6, 9, 7, 9, 8, 9, 7, 9, 6, 7, 5, 5, 3, 4, 2, 2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Emeric Deutsch, Apr 20 2007

Row n contains 1+n^2 entries. Row sums are the Catalan numbers (A000108). Column sums yield A129528. T(n,n+k) = T(n,n^2-k) (i.e., rows are palindromic). Alternating row sums are (-1)^n*binomial(n,floor(n/2)) = A126930(n). Sum_{k=0..n^2} k*T(n,k) = n*binomial(2n-1,n-1) = A002457(n-1). T(n,k) = A129175(n,n-k) (i.e., except for the initial 0's, rows of A129174 and A129175 are the same).

			T(5,11)=3 because we have (i) UDUDUUUDDD with peak-abscissae 1,3,7, (ii) UUUDDUUDD with peak-abscissae 3,8 and (iii) UUUUDDUDDD with peak-abscissae 4,7; here U=(1,1) and D=(1,-1).
Triangle starts:
  1;
  0,1;
  0,0,1,0,1;
  0,0,0,1,0,1,1,1,0,1;
  0,0,0,0,1,0,1,1,2,1,2,1,2,1,1,0,1;
  ...

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.

Alois P. Heinz, Rows n = 0..32, flattened
FindStat - Combinatorial Statistic Finder, The major index of a Dyck path.

Cf. A000108, A002457, A126930, A129175, A129528.

br:=n->sum(q^i,i=0..n-1): f:=n->product(br(j),j=1..n): cbr:=(n,k)->f(n)/f(k)/f(n-k): P:=n->sort(expand(simplify(q^n*cbr(2*n,n)/br(n+1)))): for n from 0 to 7 do seq(coeff(P(n),q,k),k=0..n^2) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
      expand(b(x-1, y+1, 1) +`if`(t=1, z^x, 1)*b(x-1, y-1, 0))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n^2))(b(2*n, 0$2)):
seq(T(n), n=0..8);  # Alois P. Heinz, Jun 10 2014

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y+1, 1] + If[t == 1, z^x, 1]*b[x-1, y-1, 0]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, z, i], {i, 0, n^2}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)

The generating polynomial for row n is P[n](t) = t^n*binomial[2n,n]/[n+1], where [n+1]=1+t+t^2+...+t^n and binomial[2n,n] is a Gaussian polynomial (in t).

A210628 Expansion of (-1 + 2x + sqrt( 1 - 4x^2)) / (2*x) in powers of x.

Original entry on oeis.org

1, -1, 0, -1, 0, -2, 0, -5, 0, -14, 0, -42, 0, -132, 0, -429, 0, -1430, 0, -4862, 0, -16796, 0, -58786, 0, -208012, 0, -742900, 0, -2674440, 0, -9694845, 0, -35357670, 0, -129644790, 0, -477638700, 0, -1767263190, 0, -6564120420, 0, -24466267020, 0

Offset: 0

Michael Somos, Mar 25 2012

sign

Except for the leading term, the sequence is equal to -A097331(n). - Fung Lam, Mar 22 2014

			G.f. = 1 - x - x^3 - 2*x^5 - 5*x^7 - 14*x^9 - 42*x^11 - 132*x^13 - 429*x^15 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A000108, A001405, A090192, A097331, A126930, A210736.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((-1 + 2*x + Sqrt(1-4*x^2))/(2*x))); // G. C. Greubel, Aug 11 2018

CoefficientList[Series[1 - 2 x/(1 + Sqrt[1 - 4 x^2]), {x, 0, 45}], x] (* Bruno Berselli, Mar 25 2012 *)
a[ n_] := SeriesCoefficient[ (-1 + 2 x + Sqrt[1 - 4 x^2]) / (2 x), {x, 0, n}];

makelist(coeff(taylor(1-2*x/(1+sqrt(1-4*x^2)), x, 0, n), x, n), n, 0, 45); /* Bruno Berselli, Mar 25 2012 */

{a(n) = polcoeff( (-1 + 2*x + sqrt( 1 - 4*x^2 + x^2 * O(x^n))) / (2*x), n)};

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

{a(n) = my(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = 1 - x - x * (1 - A)^2); polcoeff( A, n))};

G.f.: 1 - (2*x) / (1 + sqrt( 1 - 4*x^2)) = 1 - (1 - sqrt( 1 - 4*x^2)) / (2*x).
G.f. A(x) satisfies 0 = f(x, A(x)) where f(x, y) = x*y^2 - (1 - 2*x) * (1 - y).
G.f. A(x) satisfies A( x / (1 + x^2) ) = 1 - x.
G.f. A(x) = 1 - x - x * (1 - A(x))^2 = 1 - 1/x + 1 / (1 - A(x)).
G.f. A(x) = 1 / (1 + x / (1 - 2*x + x * A(x))).
G.f. A(x) = 1 / (1 + x / (1 - x / (1 - x / (1 + x * A(x))))).
G.f. A(x) = 1 / (1 + x * A001405(x)). A126930(x) = 1 / (1 + x * A(x)).
G.f. A(x) = 1 - x / (1 - x^2 / (1 - x^2 / (1 - x^2 / ...))). - Michael Somos, Jan 02 2013
a(2*n) = 0 unless n=0, a(2*n + 1) = -A000108(n). a(n) = (-1)^n * A097331(n). a(n-1) = (-1)^floor(n/2) * A090192(n).
Convolution inverse of A210736. - Michael Somos, Jan 02 2013
G.f.: 2/( G(0) + 1), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1+2*x) - 2*x*(1+2*x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1+2*x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013
D-finite with recurrence: (n+3)*a(n+2) = 4*n*a(n), a(0)=1, a(1)=-1. - Fung Lam, Mar 17 2014
For nonzero odd-power terms, a(n) = -2^(n+1)/(n+1)^(3/2)/sqrt(2*Pi)*(1+3/(4*n) + O(1/n^2)). (with contribution of Vaclav Kotesovec) - Fung Lam, Mar 17 2014

A171509 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A126931.

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 33, 29, 9, 1, 110, 126, 57, 12, 1, 366, 518, 306, 94, 15, 1, 1220, 2052, 1494, 600, 140, 18, 1, 4065, 7925, 6849, 3389, 1035, 195, 21, 1, 13550, 30030, 30025, 17628, 6635, 1638, 259, 24, 1

Offset: 0

Philippe Deléham, Dec 10 2009

Equal to A053121*B^3, B = A007318.

			Triangle begins:
  1 ;
  3,1 ;
  10,6,1 ;
  33,29,9,1 ;
  110,126,57,12,1 ; ...

Cf. A053121, A054336, A096164, A171505.

Sum_{k=0..n} T(n,k)*x^k = A126930(n), A126120(n), A001405(n), A054341(n), A126931(n) for x = -4, -3, -2, -1, 0 respectively.

T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + Sum_{i>=0} T(n-1,k+1+i)*(-3)^i. - Philippe Deléham, Feb 23 2012

cp's OEIS Frontend

A124644 Triangle read by rows. T(n, k) = binomial(n, k) * CatalanNumber(n - k).

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A129174 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n such that the sum of the peak-abscissae is k (0 <= k <= n^2).

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A210628 Expansion of (-1 + 2*x + sqrt( 1 - 4*x^2)) / (2*x) in powers of x.

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A171509 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A126931.

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A210628 Expansion of (-1 + 2x + sqrt( 1 - 4x^2)) / (2*x) in powers of x.