A100234 -id:A100234 - cp's OEIS frontend

A100235 Triangle, read by rows, of the coefficients of [x^k] in G100234(x)^n such that the row sums are 6^n-1 for n>0, where G100234(x) is the g.f. of A100234.

1, 1, 4, 1, 8, 26, 1, 12, 63, 139, 1, 16, 116, 436, 726, 1, 20, 185, 965, 2830, 3774, 1, 24, 270, 1790, 7335, 17634, 19601, 1, 28, 371, 2975, 15505, 52444, 106827, 101784, 1, 32, 488, 4584, 28860, 124424, 358748, 633952, 528526, 1, 36, 621, 6681, 49176, 256194

Offset: 0

Paul D. Hanna, Nov 29 2004

The main diagonal forms A100236. Secondary diagonal is: T(n+1,n) = (n+1)*A100237(n). More generally, if g.f. F(x) satisfies: m^n-b^n = Sum_{k=0..n} [x^k]F(x)^n, then F(x) also satisfies: (m+z)^n - (b+z)^n + z^n = Sum_{k=0..n} [x^k](F(x)+z*x)^n for all z and F(x)=(1+(m-1)*x+sqrt(1+2*(m-2*b-1)*x+(m^2-2*m+4*b+1)*x^2))/2; the triangle formed from powers of F(x) will have the g.f.: G(x,y)=(1-2*x*y+m*x^2*y^2)/((1-x*y)*(1-(m-1)*x*y-x^2*y^2-x*(1-x*y))).

			Rows begin:
[1],
[1,4],
[1,8,26],
[1,12,63,139],
[1,16,116,436,726],
[1,20,185,965,2830,3774],
[1,24,270,1790,7335,17634,19601],
[1,28,371,2975,15505,52444,106827,101784],
[1,32,488,4584,28860,124424,358748,633952,528526],...
where row sums form 6^n-1 for n>0:
6^1-1 = 1+4 = 5
6^2-1 = 1+8+26 = 35
6^3-1 = 1+12+63+139 = 215
6^4-1 = 1+16+116+436+726 = 1295
6^5-1 = 1+20+185+965+2830+3774 = 7775.
The main diagonal forms A100236 = [1,4,26,139,726,3774,...],
where Sum_{n>=1} A100236(n)/n*x^n = log((1-x)/(1-5*x-x^2)).

Cf. A100234, A100236, A100237, A100232.

row[n_] := CoefficientList[ Series[ (1 + 5*x + Sqrt[1 + 6*x + 29*x^2])^n/2^n, {x, 0, n}], x]; Flatten[ Table[ row[n], {n, 0, 9}]](* Jean-François Alcover, May 11 2012, after PARI *)

PARI
```
T(n,k,m=6)=if(n
				
```

G.f.: A(x, y)=(1-2*x*y+6*x^2*y^2)/((1-x*y)*(1-5*x*y-x^2*y^2-x*(1-x*y))).

A100237 Secondary diagonal of triangle A100235 divided by row number: a(n) = A100235(n+1,n)/(n+1) for n >= 0.

Original entry on oeis.org

1, 4, 21, 109, 566, 2939, 15261, 79244, 411481, 2136649, 11094726, 57610279, 299146121, 1553340884, 8065850541, 41882593589, 217478818486, 1129276686019, 5863862248581, 30448587928924, 158106801893201, 820982597394929, 4263019788867846, 22136081541734159

Offset: 0

Paul D. Hanna, Nov 30 2004

G.f. equals the ratio of the g.f.s of any two adjacent diagonals of triangle A100235.

a(n) is the number of compositions of n when there are 4 types of 1 and 5 types of other natural numbers. - Milan Janjic, Aug 13 2010

Colin Barker, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (5,1).

Cf. A100234, A100235, A100236.
First differences of A052918.
Cf. A052918.

a[0]:=1: a[1]:=4: for n from 2 to 26 do a[n]:=5*a[n-1]+a[n-2] od: seq(a[n], n=0..21); # Zerinvary Lajos, Jul 26 2006

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

Vec((1-x)/(1-5*x-x^2) + O(x^40)) \\ Colin Barker, Oct 13 2015

a(n) = 5*a(n-1) + a(n-2) for n>1, with a(0)=1, a(1)=4.
G.f.: (1-x)/(1-5*x-x^2).
Numerators in continued fraction [1, 4, 5, 5, 5, ...]. Continued fraction [1, 4, 5, 5, 5, ...] = 0.807417596433..., the inradius of a right triangle with legs 2 and 5. n-th convergent (n>0) to [1, 4, 5, 5, 5, ...] = A100237(n)/A052918(n), the first few being 1/1, 4/5, 21/26, 109/135, ... - Gary W. Adamson, Dec 21 2007
If p[1]=4, p[i]=5, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n >= 1, a(n)=det A. - Milan Janjic, Apr 29 2010
a(n) = (2^(-1-n)*((5-sqrt(29))^n*(-3+sqrt(29)) + (3+sqrt(29))*(5+sqrt(29))^n))/sqrt(29). - Colin Barker, Oct 13 2015

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A100235 Triangle, read by rows, of the coefficients of [x^k] in G100234(x)^n such that the row sums are 6^n-1 for n>0, where G100234(x) is the g.f. of A100234.

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A100237 Secondary diagonal of triangle A100235 divided by row number: a(n) = A100235(n+1,n)/(n+1) for n >= 0.

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