A156266 -id:A156266 - cp's OEIS frontend

A085880 Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).

1, 1, 1, 2, 4, 2, 5, 15, 15, 5, 14, 56, 84, 56, 14, 42, 210, 420, 420, 210, 42, 132, 792, 1980, 2640, 1980, 792, 132, 429, 3003, 9009, 15015, 15015, 9009, 3003, 429, 1430, 11440, 40040, 80080, 100100, 80080, 40040, 11440, 1430, 4862, 43758, 175032, 408408, 612612, 612612, 408408, 175032, 43758, 4862

Offset: 0

N. J. A. Sloane, Aug 17 2003

Coefficients of terms in the series reversion of (1-k*x-(k+1)*x^2)/(1+x). - Paul Barry, May 21 2005
Equals A131427 * A007318 as infinite lower triangular matrices. [Philippe Deléham, Sep 15 2008]
Sum_{k=0..n} T(n,k)*x^k = A168491(n), A000007(n), A000108(n), A151374(n), A005159(n), A151403(n), A156058(n), A156128(n), A156266(n), A156270(n), A156273(n), A156275(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Nov 15 2013
Diagonal sums are A052709(n+1). - Philippe Deléham, Nov 15 2013

			Triangle starts:
[ 1]     1;
[ 2]     1,     1;
[ 3]     2,     4,      2;
[ 4]     5,    15,     15,      5;
[ 5]    14,    56,     84,     56,     14;
[ 6]    42,   210,    420,    420,    210,     42;
[ 7]   132,   792,   1980,   2640,   1980,    792,    132;
[ 8]   429,  3003,   9009,  15015,  15015,   9009,   3003,    429;
[ 9]  1430, 11440,  40040,  80080, 100100,  80080,  40040,  11440,  1430;
[10]  4862, 43758, 175032, 408408, 612612, 612612, 408408, 175032, 43758, 4862;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.

Flat(List([0..10], n-> List([0..n], k-> Binomial(n,k)*Binomial(2*n,n)/( n+1) ))); # G. C. Greubel, Feb 07 2020

[Binomial(n,k)*Catalan(n): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 07 2020

seq(seq(binomial(n, k)*binomial(2*n, n)/(n+1), k = 0..n), n = 0..10); # G. C. Greubel, Feb 07 2020

Table[Binomial[n, k]*CatalanNumber[n], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 07 2020 *)

tabl(nn) = {for (n=0, nn, c =  binomial(2*n,n)/(n+1); for (k=0, n, print1(c*binomial(n, k), ", ");); print(););} \\ Michel Marcus, Apr 09 2015

[[binomial(n,k)*catalan_number(n) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 07 2020

Triangle given by [1, 1, 1, 1, 1, 1, ...] DELTA [1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
Sum_{k>=0} T(n, k) = A151374(n) (row sums). - Philippe Deléham, Aug 11 2005
G.f.: (1-sqrt(1-4*(x+y)))/(2*(x+y)). - Vladimir Kruchinin, Apr 09 2015

A290605 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 2/(1 + sqrt(1 - 4kx)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 8, 5, 0, 1, 4, 18, 40, 14, 0, 1, 5, 32, 135, 224, 42, 0, 1, 6, 50, 320, 1134, 1344, 132, 0, 1, 7, 72, 625, 3584, 10206, 8448, 429, 0, 1, 8, 98, 1080, 8750, 43008, 96228, 54912, 1430, 0, 1, 9, 128, 1715, 18144, 131250, 540672, 938223, 366080, 4862, 0

Offset: 0

Ilya Gutkovskiy, Aug 07 2017

Number of 2n-length strings of balanced parentheses of at most k different types. Also number of binary trees with n inner nodes of at most k different dimensions. - Alois P. Heinz, Oct 28 2019

			G.f. of column k: A(x) = 1 + k*x + 2*k^2*x^2 + 5*k^3*x^3 + 14*k^4*x^4 + 42*k^5*x^5 + 132*k^6*x^6 + ...
Square array begins:
  1,   1,     1,      1,      1,       1,  ...
  0,   1,     2,      3,      4,       5,  ...
  0,   2,     8,     18,     32,      50,  ...
  0,   5,    40,    135,    320,     625,  ...
  0,  14,   224,   1134,   3584,    8750,  ...
  0,  42,  1344,  10206,  43008,  131250,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give:  A000007, A000108, A151374, A005159, A151403, A156058, A156128, A156266, A156270, A156273, A156275.
Rows n=0-2 give: A000012, A001477, A001105.
Main diagonal gives A291699.
Cf. A000108, A253180, A256061.

ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
seq(seq(A(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Oct 28 2019

Table[Function[k, SeriesCoefficient[2/(1 + Sqrt[1 - 4 k x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-k x, 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

A(n,k) = k^n*(2*n)!/(n!*(n + 1)!).
A(n,k) = k^n*A000108(n).
G.f. of column k: 2/(1 + sqrt(1 - 4*k*x)).
G.f. of column k: 1/(1 - k*x/(1 - k*x/(1 - k*x/(1 - k*x/(1 - k*x/(1 - ...)))))), a continued fraction.
E.g.f. of column k: (BesselI(0,2*k*x) - BesselI(1,2*k*x))*exp(2*k*x).
If g.f. = 2/(1 + sqrt(1 - 4*k*x)), then a(n) ~ k^n*4^n/(sqrt(Pi)*n^(3/2)).
A(n,k) = Sum_{i=0..k} binomial(k,i) * A256061(n,k-i). - Alois P. Heinz, Oct 28 2019
For fixed k >= 1, Sum_{n>=0} 1/A(n,k) = 2*k*(8*k + 1) / (4*k - 1)^2 + 24 * k^2 * arcsin(1/(2*sqrt(k))) / (4*k - 1)^(5/2). - Vaclav Kotesovec, Nov 23 2021
For fixed k >= 1, Sum_{n>=0} (-1)^n / A(n,k) = 2*k*(8*k - 1) / (4*k + 1)^2 - 24 * k^2 * log((1 + sqrt(4*k + 1))/(2*sqrt(k))) / (4*k + 1)^(5/2). - Vaclav Kotesovec, Nov 24 2021

A156270 a(n) = 8^n*Catalan(n).

Original entry on oeis.org

1, 8, 128, 2560, 57344, 1376256, 34603008, 899678208, 23991418880, 652566593536, 18034567675904, 504967894925312, 14294475794808832, 408413594137395200, 11762311511156981760, 341107033823552471040, 9952299339793060331520

Offset: 0

Philippe Deléham, Feb 07 2009

A quarter of the count of And/Or-Trees with 2 variables [Chauvin]. - R. J. Mathar, Apr 01 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Brigitte Chauvin, Philippe Flajolet, Daniele Gardy and Bernhard Gittenberger, And/Or Tree Revisited, Combinat., Probal. Comput., Vol. 13, No. 4-5 (2004), pp. 475-497.

Cf. A000108, A005159, A085880, A151374, A151403, A156058, A156128, A156266.

Column k=8 of A290605.

[8^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011

Table[8^n*CatalanNumber[n], {n, 0, 20}] (* Wesley Ivan Hurt, Dec 28 2013 *)

a(n) = 8^n*A000108(n).
From Gary W. Adamson, Jul 18 2011: (Start)
a(n) is the upper left term in M^n, M = an infinite square production matrix:
  8, 8, 0, 0, 0, 0, ...
  8, 8, 8, 0, 0, 0, ...
  8, 8, 8, 8, 0, 0, ...
  8, 8, 8, 8, 8, 0, ...
  ... (End)
E.g.f.: KummerM(1/2, 2, 32*x). - Peter Luschny, Aug 26 2012
G.f.: c(8*x) with c(x) the o.g.f. of A000108 (Catalan). - Philippe Deléham, Nov 15 2013
a(n) = Sum_{k=0..n} A085880(n,k)*7^k. - Philippe Deléham, Nov 15 2013
G.f.: 1/(1 - 8*x/(1 - 8*x/(1 - 8*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Aug 08 2017
(n+1)*a(n) +16*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Apr 14 2018
Sum_{n>=0} 1/a(n) = 1040/961 + 1536*arctan(1/sqrt(31)) / (961*sqrt(31)). - Vaclav Kotesovec, Nov 23 2021
Sum_{n>=0} (-1)^n/a(n) = 112/121 - 512*arctanh(1/sqrt(33)) / (363*sqrt(33)). - Amiram Eldar, Jan 25 2022
D-finite with recurrence +(n+1)*a(n) +16*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 21 2022

A156273 a(n) = 9^n*Catalan(n).

Original entry on oeis.org

1, 9, 162, 3645, 91854, 2480058, 70150212, 2051893701, 61556811030, 1883638417518, 58564030799196, 1844766970174674, 58748732742485772, 1888352123865614100, 61182608813245896840, 1996082612532147384405, 65518476340761072970470, 2162109719245115408025510

Offset: 0

Philippe Deléham, Feb 07 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000108, A005159, A085880, A151374, A151403, A156058, A156128, A156266, A156270.

Column k=9 of A290605.

[9^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011

Table[9^n CatalanNumber[n],{n,0,20}] (* Harvey P. Dale, Sep 09 2012 *)

a(n) = 9^n*A000108(n).
From Gary W. Adamson, Jul 18 2011: (Start)
a(n) is the upper left term in M^n, M = an infinite square production matrix:
  9, 9, 0, 0, 0, 0, ...
  9, 9, 9, 0, 0, 0, ...
  9, 9, 9, 9, 0, 0, ...
  9, 9, 9, 9, 9, 0, ...
  ... (End)
E.g.f.: KummerM(1/2, 2, 36*x). - Peter Luschny, Aug 26 2012
G.f.: c(9*x) with c(x) the o.g.f. of A000108 (Catalan). - Philippe Deléham, Nov 15 2013
a(n) = Sum{k=0..n} A085880(n,k)*8^k. - Philippe Deléham, Nov 15 2013
G.f.: 1/(1 - 9*x/(1 - 9*x/(1 - 9*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Aug 08 2017
Sum_{n>=0} 1/a(n) = 1314/1225 + 1944*arctan(1/sqrt(35)) / (1225*sqrt(35)). - Vaclav Kotesovec, Nov 23 2021
Sum_{n>=0} (-1)^n/a(n) = 1278/1369 - 1944*arctanh(1/sqrt(37)) / (1369*sqrt(37)). - Amiram Eldar, Jan 25 2022
D-finite with recurrence (n+1)*a(n) +18*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 21 2022

A156362 a(2n+2) = 8a(2n+1), a(2n+1) = 8a(2n) - 7^n*A000108(n), a(0)=1.

Original entry on oeis.org

1, 7, 56, 441, 3528, 28126, 225008, 1798349, 14386792, 115060722, 920485776, 7363180314, 58905442512, 471228010428, 3769824083424, 30158239367445, 241265914939560, 1930119075851050, 15440952606808400, 123527424655229966

Offset: 0

Philippe Deléham, Feb 08 2009

nonn

Hankel transform is 7^C(n+1,2).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000108, A001405, A151162, A156195, A151254, A151281, A156266, A156361.

[n le 3 select Factorial(n+5)/720 else (8*n*Self(n-1) + 28*(n-3)*Self(n-2) - 224*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Nov 09 2022

a[n_]:= a[n]= If[n==0, 1, If[OddQ[n], 8*a[n-1] -7^((n-1)/2)*CatalanNumber[(n-1)/2], 8*a[n-1]]]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Nov 09 2022 *)

def a(n): # a = A156362
    if (n==0): return 1
    elif (n%2==1): return 8*a(n-1) - 7^((n-1)/2)*catalan_number((n-1)/2)
    else: return 8*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Nov 09 2022

a(n) = Sum_{k=0..n} A120730(n,k) * 7^k.

a(n) = ( 8*(n+1)*a(n-1) + 28*(n-2)*a(n-2) - 224*(n-2)*a(n-3) )/(n+1). - G. C. Greubel, Nov 09 2022

A156275 a(n) = 10^n*Catalan(n).

Original entry on oeis.org

1, 10, 200, 5000, 140000, 4200000, 132000000, 4290000000, 143000000000, 4862000000000, 167960000000000, 5878600000000000, 208012000000000000, 7429000000000000000, 267444000000000000000, 9694845000000000000000, 353576700000000000000000

Offset: 0

Philippe Deléham, Feb 07 2009

In general, for m >= 1, Sum_{k>=0} 1/(m^k * Catalan(k)) = 2*m*(8*m + 1) / (4*m - 1)^2 + 24 * m^2 * arcsin(1/(2*sqrt(m))) / (4*m - 1)^(5/2). - Vaclav Kotesovec, Nov 23 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vincent Pilaud, Pebble trees, arXiv:2205.06686 [math.CO], 2022.

Cf. A000108, A005159, A085880, A151374, A151403, A156058, A156128, A156266, A156270, A156273.

Column k=10 of A290605.

[10^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011

Table[10^n CatalanNumber[n],{n,0,20}] (* Harvey P. Dale, Mar 12 2013 *)

a(n) = 10^n*A000108(n).
From Gary W. Adamson, Jul 18 2011: (Start)
a(n) is the upper left term in M^n, M = an infinite square production matrix:
  10, 10,  0,  0,  0, ...
  10, 10, 10,  0,  0, ...
  10, 10, 10, 10,  0, ...
  10, 10, 10, 10, 10, ...
  ... (End)
E.g.f.: KummerM(1/2, 2, 40*x). - Peter Luschny, Aug 26 2012
G.f.: c(10*x) with c(x) the o.g.f. of A000108 (Catalan). - Philippe Deléham, Nov 15 2013
a(n) = Sum_{k=0..n} A085880(n,k)*9^k. - Philippe Deléham, Nov 15 2013
G.f.: 1/(1 - 10*x/(1 - 10*x/(1 - 10*x/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Aug 08 2017
Sum_{n>=0} 1/a(n) = 180/169 + 800*arctan(1/sqrt(39)) / (507*sqrt(39)). - Vaclav Kotesovec, Nov 23 2021
Sum_{n>=0} (-1)^n/a(n) = 1580/1681 - 2400*arctanh(1/sqrt(41)) / (1681*sqrt(41)). - Amiram Eldar, Jan 25 2022
D-finite with recurrence (n+1)*a(n) +20*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 21 2022

a(15) corrected by Vincenzo Librandi, Jul 19 2011

A112701 Partial sum of Catalan numbers (A000108) multiplied by powers of 7.

Original entry on oeis.org

1, 8, 106, 1821, 35435, 741329, 16270997, 369570944, 8613236374, 204812473608, 4949266755812, 121188396669810, 3000342229924222, 74979188061284522, 1888846103011564082, 47915719069874907917, 1222954711282739097587

Offset: 0

Wolfdieter Lang, Oct 31 2005

Robert Israel, Table of n, a(n) for n = 0..693

Column m=7 of triangle A112705.
Partial sums of A156266.
Cf. A000108, A000420.

f:= gfun:-rectoproc({(n+1)*a(n) +(-29*n+13)*a(n-1) +14*(2*n-1)*a(n-2)=0,a(0)=1,a(1)=8},a(n),remember):
map(f, [$0..50]); # Robert Israel, Aug 04 2020

CatalanNumber[#]*7^#& /@ Range[0, 20] // Accumulate (* Jean-François Alcover, Aug 29 2022 *)

a(n) = Sum_{k=0..n} A000108(k)*7^k.
G.f.: c(7*x)/(1-x), where c(x) = (1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
Conjecture: (n+1)*a(n) +(-29*n+13)*a(n-1) +14*(2*n-1)*a(n-2)=0. - R. J. Mathar, Jun 08 2016
Conjecture verified using the d.e. (28*x^3-29*x^2+x)*y' + (42*x^2-16*x+1)*y=1 satisfied by the g.f. - Robert Israel, Aug 04 2020
a(n) = 7^n*binomial(2*n, n)*(1 - hypergeom([1, n+1/2], [n+2], 28))/(n+1) + (1 - 3*sqrt(3)*i)/14, where i denotes the imaginary units. - Stefano Spezia, Mar 31 2025

cp's OEIS Frontend

A085880 Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).

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A290605 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 2/(1 + sqrt(1 - 4*k*x)).

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A156270 a(n) = 8^n*Catalan(n).

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A156273 a(n) = 9^n*Catalan(n).

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A156362 a(2*n+2) = 8*a(2*n+1), a(2*n+1) = 8*a(2*n) - 7^n*A000108(n), a(0)=1.

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A156275 a(n) = 10^n*Catalan(n).

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A112701 Partial sum of Catalan numbers (A000108) multiplied by powers of 7.

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A290605 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 2/(1 + sqrt(1 - 4kx)).

A156362 a(2n+2) = 8a(2n+1), a(2n+1) = 8a(2n) - 7^n*A000108(n), a(0)=1.