A016196 -id:A016196 - cp's OEIS frontend

A102728 Array read by antidiagonals: T(n, k) = ((n+1)^k-(n-1)^k)/2.

0, 0, 1, 0, 1, 0, 0, 1, 2, 1, 0, 1, 4, 4, 0, 0, 1, 6, 13, 8, 1, 0, 1, 8, 28, 40, 16, 0, 0, 1, 10, 49, 120, 121, 32, 1, 0, 1, 12, 76, 272, 496, 364, 64, 0, 0, 1, 14, 109, 520, 1441, 2016, 1093, 128, 1, 0, 1, 16, 148, 888, 3376, 7448, 8128, 3280, 256, 0, 0, 1, 18, 193, 1400, 6841

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 07 2005

Consider a 2 X 2 matrix M = [N, 1] / [1, N]. The n-th row of the array contains the values of the non-diagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = non-diagonal entry + (N-1)^k.) Table:
N: row sequence g.f. cross references.
(1^n-(-1)^n)/2 x/((1+1x)(1-1x)) A000035
(2^n-0^n)/2 x/(1-2x) A000079
(3^n-1^n)/2 x/((1-1x)(1-3x)) A003462
(4^n-2^n)/2 x/((1-2x)(1-4x)) A006516
(7^n-3^n)/2 x/((1-3x)(1-5x)) A005059
(6^n-4^n)/2 x/((1-4x)(1-6x)) A016149
(7^n-5^n)/2 x/((1-5x)(1-7x)) A016161 A081200
(8^n-6^n)/2 x/((1-6x)(1-8x)) A016170 A081201
(9^n-7^n)/2 x/((1-7x)(1-9x)) A016178 A081202
(10^n-8^n)/2 x/((1-8x)(1-10x)) A016186 A081203
(11^n-9^n)/2 x/((1-9x)(1-11x)) A016190
(12^n-10^n)/2 x/((1-10x)(1-12x)) A016196
...
Characteristic polynomial x^2-2nx+(n^2-1) has roots n+-1, so if r(n) denotes a row sequence, r(n+1)/r(n) converges to n+1.
Columns follow polynomials with certain binomial coefficients:
column: polynomial
0
1
2n
3n^2+ 1 (see A056107)
4n^3+ 4n (= 8*A006003(n))
5n^4+ 10n^2+ 1
6n^5+ 20n^3+ 6n
7n^6+ 35n^4+ 21n^2+ 1
8; 8n^7+ 56n^5+ 56n^3+ 8n
9n^8+ 84n^6+126n^4+ 36n^2+ 1
10n^9+ 120n^7+252n^5+120n^3+ 10n
11n^10+165n^8+462n^6+330n^4+ 55n^2+ 1

			Array begins:
0,1,0,1,0,1...
0,1,2,4,8,16...
0,1,4,13,40,121...
0,1,6,28,120,496...
0,1,8,49,272,1441...
...

MM(n,N)=local(M);M=matrix(n,n);for(i=1,n, for(j=1,n,if(i==j,M[i,j]=N,M[i,j]=1)));M for(k=0,12, for(i=0,k,print1((MM(2,k-i)^i)[1,2],","))) T(n, k) = ((n+1)^k-(n-1)^k)/2 for(k=0,10, for(i=0,10,print1(T(k,i),","));print()) for(k=0,10, for(i=0,10,print1(((k+1)^i-(k-1)^i)/2,","));print()) for(k=0,10, for(i=0,10,print1(polcoeff(x/((1-(k-1)*x)*(1-(k+1)*x)),i),","));print())

A021724 Expansion of 1/((1-x)(1-3x)(1-10x)(1-12x)).

Original entry on oeis.org

1, 26, 465, 7150, 101621, 1378026, 18123145, 233349350, 2958918141, 37094306626, 461004657425, 5690785933950, 69876732453061, 854393804284826, 10411455807073305, 126524771262956950, 1534170271000826381

Offset: 0

N. J. A. Sloane

From Bruno Berselli, May 08 2013: (Start)
Naturally, the sequence is related to:
A018207, 1/((1-3x)(1-10x)(1-12x)):  A018207(n) = a(n)-a(n-1), n>0;
A016267, 1/((1-x)(1-10x)(1-12x)):  A016267(n) = a(n)-3*a(n-1), n>0;
A016217, 1/((1-x)(1-3x)(1-12x)):  A016217(n) = a(n)-10*a(n-1), n>0;
A016215, 1/((1-x)(1-3x)(1-10x)):  A016215(n) = a(n)-12*a(n-1), n>0;
A016196, 1/((1-10x)(1-12x)):  A016196(n) = a(n)-4*a(n-1)+3*a(n-2), n>1;
A016147, 1/((1-3x)(1-12x)):  A016147(n) = a(n)-11*a(n-1)+10*a(n-2), n>1;
A016145, 1/((1-3x)(1-10x)):  A016145(n) = a(n)-13*a(n-1)+12*a(n-2), n>1;
A016125, 1/((1-x)(1-12x)):  A016125(n) = a(n)-13*a(n-1)+30*a(n-2), n>1;
A002275, x/((1-x)(1-10x)):  A002275(n) = a(n-1)-15*a(n-2)+36*a(n-3), n>2;
A003462, x/((1-x)(1-3x)):  A003462(n) = a(n-1)-22*a(n-2)+120*a(n-3), n>2;
A000012, 1/(1-x):  A000012(n) = a(n)-25*a(n-1)+186*a(n-2)-360*a(n-3), n>2;
A000244, 1/(1-3x):  A000244(n) = a(n)-23*a(n-1)+142*a(n-2)-120*a(n-3), n>2;
A011557, 1/(1-10x):  A011557(n) = a(n)-16*a(n-1)+51*a(n-2)-36*a(n-3), n>2;
A001021, 1/(1-12x):  A001021(n) = a(n)-14*a(n-1)+43*a(n-2)-30*a(n-3), n>2. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (26,-211,546,-360).

Cf. A000012, A000244, A001021, A002275, A003462, A011557, A016125, A016145, A016147, A016196, A016215, A016217, A016267, A018207.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x)))); // Bruno Berselli, May 07 2013

CoefficientList[Series[1/((1 - x) (1 - 3 x) (1 - 10 x) (1 - 12 x)), {x, 0, 20}], x] (* Bruno Berselli, May 07 2013 *)
LinearRecurrence[{26,-211,546,-360},{1,26,465,7150},120] (* Harvey P. Dale, Jul 06 2019 *)

Vec(1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x))+O(x^20)) \\ Bruno Berselli, May 07 2013

G.f.: 1/((1-x)*(1-3*x)*(1-10*x)*(1-12*x)).

a(n) = -1/198 +3^(n+1)/14 -2^(n+2)*5^(n+3)/63 +2^(2n+5)*3^(n+1)/11. [Bruno Berselli, May 07 2013]

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A102728 Array read by antidiagonals: T(n, k) = ((n+1)^k-(n-1)^k)/2.

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A021724 Expansion of 1/((1-x)(1-3x)(1-10x)(1-12x)).

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