A081696 -id:A081696 - cp's OEIS frontend

A205813 Triangle T(n,k), read by rows, given by (0, 2, 1, 1, 1, 1, 1, 1, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 20, 16, 6, 1, 0, 70, 64, 30, 8, 1, 0, 252, 256, 140, 48, 10, 1, 0, 924, 1024, 630, 256, 70, 12, 1, 0, 3432, 4096, 2772, 1280, 420, 96, 14, 1, 0, 12870, 16384, 12012, 6144, 2310, 640, 126, 16, 1

Offset: 0

Philippe Deléham, Feb 01 2012

Riordan array (1, x/sqrt(1-4*x)). Inverse of Riordan array (1, x*exp(arcsinh(-2*x))).

T is the convolution triangle of the shifted central binomial coefficients binomial(2*(n-1), n-1). - Peter Luschny, Oct 19 2022

			Triangle begins:
  1;
  0,   1;
  0,   2,   1;
  0,   6,   4,   1;
  0,  20,  16,   6,   1;
  0,  70,  64,  30,   8,   1;
  0, 252, 256, 140,  48,  10,   1;

Cf. A054335 and columns listed there.

# Uses function PMatrix from A357368.
PMatrix(10, n -> binomial(2*(n-1), n-1)); # Peter Luschny, Oct 19 2022

T(n,n) = 1 = A000012(n); T(n+1,n) = 2*n = A005843(n); T(n+2,n) = 2*n*(n+2) = A054000(n+1).
Sum_{k=0..n} T(n,k)*x^k = -A081696(n-1), A000007(n), A026671(n-1), A084868(n) for x = -1, 0, 1, 2 respectively.
G.f.: sqrt(1-4*x)/(sqrt(1-4*x)-y*x).
Sum_{k=0..n} T(n,k)*A090192(k) = A000108(n), A000108 = Catalan numbers.

A081698 Expansion of (1 - sqrt( 1 - 4xsqrt( 1 + 4x )) )/( 2x ).

Original entry on oeis.org

1, 3, 4, 21, 56, 282, 984, 4813, 19280, 93150, 403672, 1945954, 8845360, 42766292, 200419504, 974134461, 4659558048, 22785183670, 110564976792, 543935554390, 2667398588272, 13196971915628, 65238895435792, 324431740601618, 1614044041864800, 8063536826420460

Offset: 0

Emanuele Munarini, Apr 02 2003

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A081696.

a:= proc(n) option remember; `if`(n<4, [1, 3, 4, 21][n+1],
      (2*n*(n+1)*(3-2*n) *a(n-1) +4*n*(2*n-1)*(2*n-3) *a(n-2)
       +8*(2*n-3)*(8*n^2-16*n-15) *a(n-3)
       +16*(4*n-15)*(4*n-9)*(n+1) *a(n-4)) /(n^2*(n+1)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 13 2013

a[n_] := Sum[Binomial[(k+1)/2, n-k]*Binomial[2*k, k]*4^(n-k)/(k+1), {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 02 2015, after Vladimir Kruchinin *)
CoefficientList[Series[(1-Sqrt[1-4x Sqrt[1+4x]])/(2x),{x,0,30}],x] (* Harvey P. Dale, Oct 30 2017 *)

G.f.: (1-sqrt(1-4*x*sqrt(1+4*x)))/(2*x).
a(n) = sum(k=0..n, (binomial((k+1)/2,n-k)*binomial(2*k,k)*4^(n-k))/(k+1)). [Vladimir Kruchinin, Mar 13 2013]
D-finite with recurrence: n*(n+1)*a(n) +2*n*(5*n-7)*a(n-1) +4*(2*n^2-13*n+12)*a(n-2) -8*(2*n-3)*(14*n-37)*a(n-3) +16*(-64*n^2+392*n-573)*a(n-4) -96*(4*n-13)*(4*n-19)*a(n-5)=0. - R. J. Mathar, Jan 23 2020

A152229 Eigentriangle, row sums = A000984.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 9, 3, 2, 6, 29, 9, 6, 6, 20, 97, 29, 18, 18, 20, 70, 333, 97, 58, 54, 60, 70, 252, 1165, 333, 194, 174, 180, 210, 252, 924, 4135, 1165, 666, 582, 580, 630, 756, 924, 3432, 14845, 4135, 2330, 1998, 1940, 2030, 2268, 2772, 3432, 12870

Offset: 0

Gary W. Adamson, Nov 29 2008

Row sums = A000984: (1, 2, 6, 20, 70, 252,...), left border = A081696.

Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
1, 1;
3, 1, 2;
9, 3, 2, 6;
29, 9, 6, 6, 20;
97, 29, 18, 18, 20, 70;
333, 97, 58, 54, 60, 70, 252;
1165, 333, 194, 174, 180, 210, 252, 924;
4135, 1165, 666, 582, 580, 630, 756, 924, 3432;
14845, 4135, 2330, 1998, 1940, 2030, 2268, 2772, 3432, 12870;
...
Row 3 = (9, 3, 2, 6) = termwise products of (9, 3, 1, 1) and (1, 1, 2, 6).

Cf. A000984, A081696.

Triangle read by rows, M*Q. M = an infinite lower triangular matrix with A081696: (1, 1, 3, 9, 29, 97, 333, 1165,...) in every column; and Q = a matrix with A000984 as the main diagonal (prefaced with a 1): (1, 1, 2, 6, 20, 70, 252,...) and the rest zeros.

A155788 Renewal array for 1/(x+sqrt(1-4x)).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 9, 7, 3, 1, 29, 24, 12, 4, 1, 97, 85, 46, 18, 5, 1, 333, 306, 177, 76, 25, 6, 1, 1165, 1115, 681, 315, 115, 33, 7, 1, 4135, 4100, 2622, 1288, 510, 164, 42, 8, 1, 14845, 15185, 10104, 5220, 2206, 774, 224, 52, 9, 1

Offset: 0

Paul Barry, Jan 27 2009

First column is A081696. Row sums are A000984.
Contribution from Paul Barry, Jan 27 2009: (Start)
First column of A155788.
In general, the image of the sequence with g.f. 1/(1-ax-bx^2) under (1,xc(x)) has g.f.
1/(1-ax-(a+b)x/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-..... (continued fraction). (End)

			Triangle begins
1,
1, 1,
3, 2, 1,
9, 7, 3, 1,
29, 24, 12, 4, 1,
97, 85, 46, 18, 5, 1,
333, 306, 177, 76, 25, 6, 1

Riordan array (1/(x+sqrt(1-4x)),x/(x+sqrt(1-4x));
G.f.: 1/(1-x-xy-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-..... (continued fraction).
G.f: 1/(1-x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-.... (continued fraction). [From Paul Barry, Jan 27 2009]

A188513 Riordan matrix (1/(x+sqrt(1-4x)),(1-sqrt(1-4x))/(2(x+sqrt(1-4x)))).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 9, 11, 5, 1, 29, 40, 23, 7, 1, 97, 147, 99, 39, 9, 1, 333, 544, 413, 194, 59, 11, 1, 1165, 2025, 1691, 907, 333, 83, 13, 1, 4135, 7575, 6842, 4078, 1725, 524, 111, 15, 1, 14845, 28455, 27464, 17856, 8453, 2979, 775, 143, 17, 1, 53791, 107277, 109631, 76718, 39851, 15804, 4797, 1094, 179, 19, 1

Offset: 0

Emanuele Munarini, Apr 02 2011

First column = sequence A081696

Row sums = sequence A101850

			Triangle begins:
  1
  1, 1
  3, 3, 1
  9, 11, 5, 1
  29, 40, 23, 7, 1
  97, 147, 99, 39, 9, 1
  333, 544, 413, 194, 59, 11, 1
  1165, 2025, 1691, 907, 333, 83, 13, 1
  4135, 7575, 6842, 4078, 1725, 524, 111, 15, 1

Cf. A081696, A101850.

Flatten[Table[Sum[Binomial[i+k,k]Binomial[2n-i,n+k+i](2k+3i+1)/(n+k+i+1),{i,0,Floor[(n-k)/2]}],{n,0,10},{k,0,n}]]

create_list(sum(binomial(i+k,k)*binomial(2*n-i,n+k+i)*(2*k+3*i+1)/(n+k+i+1),i,0,floor((n-k)/2)),n,0,10,k,0,n);

T(n,k) = [x^n] ((1-sqrt(1-4*x))/(2*(x+sqrt(1-4*x))))^k/(x+sqrt(1-4*x)).
T(n,k) = [x^(n-k)] (1-2*x)/((1-x)^(n+1)*(1-x-x^2)^(k+1)).
T(n,k) = sum(binomial(i+k,k)*binomial(2*n-i,n+k+i)*(2*k+3*i+1)/(n+k+i+1), i=0..floor((n-k)/2)).

A189675 Composition of Catalan and Fibonacci numbers.

Original entry on oeis.org

1, -1, 2, 2, -4, 3, -5, 10, -9, 5, 14, -28, 27, -20, 8, -42, 84, -84, 70, -40, 13, 132, -264, 270, -240, 160, -78, 21, -429, 858, -891, 825, -600, 351, -147, 34, 1430, -2860, 3003, -2860, 2200, -1430, 735, -272, 55, -4862, 9724, -10296, 10010, -8008, 5577, -3234, 1496, -495, 89, 16796, -33592, 35802, -35360, 29120, -21294, 13377, -7072, 2970, -890, 144, -58786, 117572, -125970, 125970, -106080, 80444, -53508, 30940, -15015, 5785, -1584, 233

Offset: 1

Wouter Meeussen, Apr 25 2011

Row sums equal 1 (proof by Bill Gosper, Apr 17 2011). Row sums of absolute terms equal A081696.

			Table starts
   1,
  -1,  2,
   2, -4,  3,
  -5, 10, -9, 5,

Email of R. W. Gosper on the math-fun mailing list, Apr 17 2011.

Cf. A081696, A171567, A064310.

Table[(-1)^(k + n) k/(2n - k) Binomial[2n - k, n - k] Fibonacci[k + 1], {n, 12}, {k, n}]

A100240 G.f. A(x) satisfies: 4^n/2 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: ((4+z)^n + z^n)/2 = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.

Original entry on oeis.org

1, 1, 2, 2, 0, -4, -6, 2, 22, 30, -26, -154, -172, 288, 1190, 990, -3040, -9620, -4970, 31350, 79120, 12580, -318210, -649610, 174150, 3185686, 5233514, -4273078, -31452228, -40495600, 64593386, 305819154, 290278982, -835918098, -2921409370, -1771072346, 9995237616, 27317409988

Offset: 0

Paul D. Hanna, Nov 30 2004

sign

			From the table of powers of A(x), we see that
4^n/2 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1=[1,1],2,2,0,-4,-6,2,22,30,-26,...
A^2=[1,2,5],8,8,0,-16,-24,8,88,120,...
A^3=[1,3,9,19],30,30,2,-54,-84,20,288,...
A^4=[1,4,14,36,73],112,112,16,-176,-288,32,...
A^5=[1,5,20,60,145,281],420,420,90,-570,-988,...
A^6=[1,6,27,92,255,582,1085],1584,1584,440,-1848,...
A^7=[1,7,35,133,413,1071,2331,4201],6006,6006,2002,...
A^8=[1,8,44,184,630,1816,4460,9320,16305],22880,22880,...
the main diagonal of which is:
[x^n]A(x)^(n+1) = (n+1)*A081696(n) for n>=0.

Cf. A100223, A057083, A007440, A081696.

CoefficientList[Series[2*x + Sqrt[1 - 2*x + 5*x^2], {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 07 2021 *)

a(n)=if(n==0,1,(4^n/2-sum(k=0,n,polcoeff(sum(j=0,min(k,n-1),a(j)*x^j)^n+x*O(x^k),k)))/n)

a(n)=polcoeff(2*x+sqrt(1-2*x+5*x^2+x^2*O(x^n)),n)

G.f.: A(x) = 2*x+sqrt(1-2*x+5*x^2).

Recurrence: n*a(n) = (2*n-3)*a(n-1) - 5*(n-3)*a(n-2). - Vaclav Kotesovec, Feb 07 2021

cp's OEIS Frontend

A205813 Triangle T(n,k), read by rows, given by (0, 2, 1, 1, 1, 1, 1, 1, 1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A081698 Expansion of (1 - sqrt( 1 - 4*x*sqrt( 1 + 4*x )) )/( 2*x ).

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A152229 Eigentriangle, row sums = A000984.

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A155788 Renewal array for 1/(x+sqrt(1-4x)).

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A188513 Riordan matrix (1/(x+sqrt(1-4x)),(1-sqrt(1-4x))/(2(x+sqrt(1-4x)))).

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Maxima

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A189675 Composition of Catalan and Fibonacci numbers.

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A100240 G.f. A(x) satisfies: 4^n/2 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: ((4+z)^n + z^n)/2 = Sum_{k=0..n} [x^k](A(x)+z*x)^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.

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A081698 Expansion of (1 - sqrt( 1 - 4xsqrt( 1 + 4x )) )/( 2x ).