Original entry on oeis.org
1, 2, 10, 35, 118, 392, 1297, 4286, 14158, 46763, 154450, 510116, 1684801, 5564522, 18378370, 60699635, 200477278, 662131472, 2186871697, 7222746566, 23855111398, 78788080763, 260219353690, 859446141836, 2838557779201
Offset: 0
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LinearRecurrence[{4,-2,-1},{1,2,10},30] (* Harvey P. Dale, May 06 2012 *)
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a(n)=if(n==0,1,n*polcoeff(log((1-x)/(1-3*x-x^2)+x*O(x^n)),n))
A100228
G.f. A(x) satisfies: 4^n - 1 = Sum_{k=0..n} [x^k] A(x)^n and also satisfies: (4+z)^n - (1+z)^n + z^n = 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, 2, 3, -3, -6, 24, 3, -183, 273, 1131, -4407, -3081, 48360, -54750, -396195, 1282551, 1860186, -17122944, 11240049, 166745823, -432682314, -1054472016, 6822994737, -835915197, -76044224139, 152526011235, 587055710271, -2871405804783, -1378878506592, 36081844133766
Offset: 0
From the table of powers of A(x) (A100229), we see that
4^n-1 = Sum of coefficients [x^0] through [x^n] in A(x)^n:
A^1=[1,2],3,-3,-6,24,3,-183,273,...
A^2=[1,4,10],6,-15,6,75,-174,-276,...
A^3=[1,6,21,35],9,-36,63,72,-612,...
A^4=[1,8,36,92,118],12,-66,192,-147,...
A^5=[1,10,55,185,380,392],15,-105,420,...
A^6=[1,12,78,322,879,1506,1297],18,-153,...
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CoefficientList[Series[(1+3x+Sqrt[1+2x+13x^2])/2,{x,0,30}],x] (* or *) Join[{1},RecurrenceTable[{a[1]==2,a[2]==3,a[n]==-((2n-3)a[n-1]+ 13(n-3)a[n-2])/n},a,{n,30}]] (* Harvey P. Dale, Feb 29 2012 *)
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{a(n) = if(n==0,1,(4^n-1-sum(k=0,n,polcoeff(sum(j=0,min(k,n-1),a(j)*x^j)^n+x*O(x^k),k)))/n)}
for(n=0,20,print1(a(n),", "))
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{a(n) = if(n==0,1,if(n==1,2,if(n==2,3,-((2*n-3)*a(n-1)+13*(n-3)*a(n-2))/n)))}
for(n=0,30,print1(a(n),", "))
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{a(n) = polcoeff((1+3*x+sqrt(1+2*x+13*x^2+x^2*O(x^n)))/2,n)}
for(n=0,30,print1(a(n),", "))
A100232
Triangle, read by rows, of the coefficients of [x^k] in G100231(x)^n such that the row sums are 5^n-1 for n>0, where G100231(x) is the g.f. of A100231.
Original entry on oeis.org
1, 1, 3, 1, 6, 17, 1, 9, 39, 75, 1, 12, 70, 220, 321, 1, 15, 110, 470, 1165, 1363, 1, 18, 159, 852, 2895, 5922, 5777, 1, 21, 217, 1393, 5943, 16807, 29267, 24475, 1, 24, 284, 2120, 10822, 38536, 93468, 141688, 103681, 1, 27, 360, 3060, 18126, 77274, 236748
Offset: 0
Rows begin:
[1],
[1,3],
[1,6,17],
[1,9,39,75],
[1,12,70,220,321],
[1,15,110,470,1165,1363],
[1,18,159,852,2895,5922,5777],
[1,21,217,1393,5943,16807,29267,24475],
[1,24,284,2120,10822,38536,93468,141688,103681],...
where row sums form 5^n-1 for n>0:
5^1-1 = 1+3 = 4
5^2-1 = 1+6+17 = 24
5^3-1 = 1+9+39+75 = 124
5^4-1 = 1+12+70+220+321 = 624
5^5-1 = 1+15+110+470+1165+1363 = 3124.
The main diagonal forms A100233 = [1,3,17,75,321,1363,5777,...],
where Sum_{n>=1} A100233(n)/n*x^n = log((1-x)/(1-4*x-x^2)).
Showing 1-3 of 3 results.
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