A085880
Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).
Original entry on oeis.org
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
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;
...
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Flat(List([0..10], n-> List([0..n], k-> Binomial(n,k)*Binomial(2*n,n)/( n+1) ))); # G. C. Greubel, Feb 07 2020
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[Binomial(n,k)*Catalan(n): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 07 2020
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seq(seq(binomial(n, k)*binomial(2*n, n)/(n+1), k = 0..n), n = 0..10); # G. C. Greubel, Feb 07 2020
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Table[Binomial[n, k]*CatalanNumber[n], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 07 2020 *)
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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
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[[binomial(n,k)*catalan_number(n) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 07 2020
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)).
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
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, ...
Columns k=0-10 give:
A000007,
A000108,
A151374,
A005159,
A151403,
A156058,
A156128,
A156266,
A156270,
A156273,
A156275.
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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
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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
A156128
a(n) = 6^n * Catalan(n).
Original entry on oeis.org
1, 6, 72, 1080, 18144, 326592, 6158592, 120092544, 2401850880, 48997757952, 1015589892096, 21327387734016, 452796847276032, 9702789584486400, 209580255024906240, 4558370546791710720, 99747873141559787520, 2194453209114315325440, 48508965675158549299200
Offset: 0
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[6^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011
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A156128_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 6*(a[w-1]+add(a[j]*a[w-j-1],j=1..w-1)) od;convert(a,list)end: A156128_list(16); # Peter Luschny, May 19 2011
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Table[CatalanNumber[n]6^n, {n, 0, 16}] (* Alonso del Arte, Jul 19 2011 *)
A156266
a(n) = 7^n*Catalan(n).
Original entry on oeis.org
1, 7, 98, 1715, 33614, 705894, 15529668, 353299947, 8243665430, 196199237234, 4744454282204, 116239129913998, 2879153833254412, 71978845831360300, 1813866914950279560, 46026872966863343835, 1175038992212864189670
Offset: 0
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[7^n*Catalan(n): n in [0..20]]; // Vincenzo Librandi, Jul 19 2011
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A156266_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 7*(a[w-1]+add(a[j]*a[w-j-1],j=1..w-1)) od;convert(a,list)end: A156266_list(16); # Peter Luschny, May 19 2011
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Table[7^n * CatalanNumber[n], {n, 0, 16}] (* Amiram Eldar, Jan 25 2022 *)
A151254
Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, 0, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}.
Original entry on oeis.org
1, 4, 20, 96, 480, 2368, 11840, 58880, 294400, 1468416, 7342080, 36667392, 183336960, 916144128, 4580720640, 22896574464, 114482872320, 572320645120, 2861603225600, 14306741583872, 71533707919360, 357650927714304, 1788254638571520, 8941026626502656, 44705133132513280, 223522175800311808
Offset: 0
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[n le 3 select Factorial(n+2)/6 else (5*n*Self(n-1) + 16*(n-3)*Self(n-2) - 80*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Nov 09 2022
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aux[i_, j_, k_, n_]:= Which[Min[i, j, k, n]<0 || Max[i, j, k]>n, 0, n==0, KroneckerDelta[i, j, k, n], True, aux[i, j, k, n] = aux[-1+i, -1+j, -1+k, -1+n] + aux[-1+i, -1+j, k, -1+n] + aux[-1+i, j, -1+k, -1+n] + aux[-1+i, j, k, -1 + n] + aux[1+i, j, k, -1+n]]; Table[Sum[aux[i,j,k,n], {i,0,n}, {j,0,n}, {k,0,n}], {n, 0, 30}]
a[n_]:= a[n]= If[n<3, (n+3)!/3!, (5*(n+1)*a[n-1] +16*(n-2)*a[n-2] -80*(n-2)*a[n- 3])/(n+1)]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Nov 09 2022 *)
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def a(n): # a = A151254
if (n==0): return 1
elif (n%2==1): return 5*a(n-1) - 4^((n-1)/2)*catalan_number((n-1)/2)
else: return 5*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Nov 09 2022
A156195
a(2n+2) = 6*a(2n+1), a(2n+1) = 6*a(2n) - 5^n*A000108(n), a(0)=1.
Original entry on oeis.org
1, 5, 30, 175, 1050, 6250, 37500, 224375, 1346250, 8068750, 48412500, 290343750, 1742062500, 10450312500, 62701875000, 376177734375, 2257066406250, 13541839843750, 81251039062500, 487496738281250, 2924980429687500, 17549718554687500, 105298311328125000
Offset: 0
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[n le 3 select Factorial(n+3)/24 else (6*n*Self(n-1) + 20*(n-3)*Self(n-2) - 120*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Nov 09 2022
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A156195 := proc(n)
option remember;
local nh;
if n= 0 then
1;
elif type(n,'even') then
6*procname(n-1);
else
nh := floor(n/2) ;
6*procname(n-1)-5^nh*A000108(nh) ;
end if;
end proc: # R. J. Mathar, Jul 21 2016
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CoefficientList[Series[(Sqrt[1-20x^2]+10x-1)/(10x(1-6x)),{x,0,30}],x] (* Harvey P. Dale, Oct 21 2016 *)
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def a(n): # a = A156195
if (n==0): return 1
elif (n%2==1): return 6*a(n-1) - 5^((n-1)/2)*catalan_number((n-1)/2)
else: return 6*a(n-1)
[a(n) for n in (0..30)] # G. C. Greubel, Nov 09 2022
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
- 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.
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
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
A354736
a(0) = a(1) = 1; a(n) = 5 * Sum_{k=0..n-2} a(k) * a(n-k-2).
Original entry on oeis.org
1, 1, 5, 10, 55, 150, 775, 2550, 12500, 46250, 219375, 875000, 4075000, 17071250, 78796875, 341100000, 1569350000, 6947531250, 31966000000, 143761750000, 662668906250, 3014440000000, 13932834296875, 63921914062500, 296358191406250, 1368603488281250, 6365085546875000
Offset: 0
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a[0] = a[1] = 1; a[n_] := a[n] = 5 Sum[a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 26}]
nmax = 26; CoefficientList[Series[(1 - Sqrt[1 - 20 x^2 (1 + x)])/(10 x^2), {x, 0, nmax}], x]
Showing 1-10 of 10 results.
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