Original entry on oeis.org
1, 17, 178, 1477, 10654, 69930, 428772, 2496813, 13962982, 75582078, 398302268, 2052354850, 10375356460, 51596749300, 252953904072, 1224672639357, 5863899363510, 27801377704310, 130648178243660, 609082400931158
Offset: 0
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m:=20; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-Sqrt(1-4*x))/(2*x*(1-4*x)^4) )); // G. C. Greubel, Feb 17 2019
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CoefficientList[Series[(1-Sqrt[1-4*x])/(2*x*(1-4*x)^4), {x, 0, 20}], x] (* G. C. Greubel, Feb 17 2019 *)
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my(x='x+O('x^20)); Vec((1-sqrt(1-4*x))/(2*x*(1-4*x)^4)) \\ G. C. Greubel, Feb 17 2019
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((1-sqrt(1-4*x))/(2*x*(1-4*x)^4)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Feb 17 2019
Original entry on oeis.org
1, 15, 142, 1083, 7266, 44758, 259356, 1435347, 7663898, 39761282, 201483204, 1001098462, 4891910100, 23565178380, 112118316088, 527674017411, 2459747256138, 11368724035210, 52145629874100, 237541552456362
Offset: 0
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[(Binomial(n+5,4)*Catalan(n+4) -5*4^(n+1)*Binomial(n+3,2))/10: n in [0..40]]; // G. C. Greubel, Jul 19 2024
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Table[(Binomial[n+5,4]*CatalanNumber[n+4] -5*4^(n+1)*Binomial[n+3,2] )/10, {n,0,40}] (* G. C. Greubel, Jul 19 2024 *)
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[(binomial(n+5,4)*catalan_number(n+4) - 5*4^(n+1)*binomial(n+3,2))/10 for n in range(41)] # G. C. Greubel, Jul 19 2024
A046527
A triangle related to A000108 (Catalan) and A000302 (powers of 4).
Original entry on oeis.org
1, 1, 1, 2, 5, 1, 5, 22, 9, 1, 14, 93, 58, 13, 1, 42, 386, 325, 110, 17, 1, 132, 1586, 1686, 765, 178, 21, 1, 429, 6476, 8330, 4746, 1477, 262, 25, 1, 1430, 26333, 39796, 27314, 10654, 2525, 362, 29, 1, 4862, 106762, 185517, 149052, 69930, 20754, 3973, 478, 33, 1
Offset: 0
Triangle begins as:
1;
1, 1;
2, 5, 1;
5, 22, 9, 1;
14, 93, 58, 13, 1;
42, 386, 325, 110, 17, 1;
132, 1586, 1686, 765, 178, 21, 1;
429, 6476, 8330, 4746, 1477, 262, 25, 1;
1430, 26333, 39796, 27314, 10654, 2525, 362, 29, 1;
4862, 106762, 185517, 149052, 69930, 20754, 3973, 478, 33, 1;
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A046527:= func< n,k | k eq 0 select Catalan(n) else (1/2)*Binomial(n, k-1)*(4^(n-k+1) - Binomial(2*n, n)/(k*Catalan(k-1))) >;
[A046527(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 28 2024
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T[n_, k_]:= If[k==0, CatalanNumber[n], (1/2)*Binomial[n,k-1]*(4^(n-k+ 1) -Binomial[2*n,n]/Binomial[2*(k-1),k-1])];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 28 2024 *)
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def A046527(n,k):
if k==0: return catalan_number(n)
else: return (1/2)*binomial(n, k-1)*(4^(n-k+1) - binomial(2*n, n)/binomial(2*(k-1), k-1))
flatten([[A046527(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 28 2024
A090299
Table T(n,k), n>=0 and k>=0, read by antidiagonals : the k-th column given by the k-th polynomial K_k related to A090285.
Original entry on oeis.org
1, 1, 1, 2, 3, 1, 5, 10, 5, 1, 14, 35, 22, 7, 1, 42, 126, 93, 38, 9, 1, 132, 462, 386, 187, 58, 11, 1, 429, 1716, 1586, 874, 325, 82, 13, 1, 1430, 6435, 6476, 3958, 1686, 515, 110, 15, 1, 4862, 24310, 26333, 17548, 8330, 2934, 765, 142, 17, 1
Offset: 0
row n=0 : 1, 1, 2, 5, 14, 42, 132, 429, ... see A000108.
row n=1 : 1, 3, 10, 35, 126, 462, 1716, 6435, ... see A001700.
row n=2 : 1, 5, 22, 93, 386, 1586, 6476, ... see A000346.
row n=3 : 1, 7, 38, 187, 874, 3958, 17548, ... see A000531.
row n=4 : 1, 9, 58, 325, 1686, 8330, 39796, ... see A018218.
Other rows :
A029887,
A042941,
A045724,
A042985,
A045492. Columns :
A000012,
A005408. Row n is the convolution of the row (n-j) with
A000984,
A000302,
A002457,
A002697 (first term omitted),
A002802,
A038845,
A020918,
A038846,
A020920 for j=1, 2, ..9 respectively.
Corrected by Alford Arnold, Oct 18 2006
Showing 1-4 of 4 results.
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