A201780 -id:A201780 - cp's OEIS frontend

A052156 Number of compositions of n into 2*j-1 kinds of j's for all j>=1.

1, 1, 4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196, 708588, 2125764, 6377292, 19131876, 57395628, 172186884, 516560652, 1549681956, 4649045868, 13947137604, 41841412812, 125524238436, 376572715308, 1129718145924

Offset: 0

Barry E. Williams, Jan 24 2000

First differences of A025192, also second differences of A000244.

			1 + x + 4*x^2 + 12*x^3 + 36*x^4 + 108*x^5 + 324*x^6 + 972*x^7 + 2916*x^8 + ...

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.
P. Ribenhoim, The Little Book of Big Primes, Springer-Verlag, N.Y., 1991, p. 53.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3).

Cf. A025192, A000244, A003462.

CoefficientList[Series[(1 - x)^2/(1 - 3 x), {x, 0, 40}], x ] (* Vincenzo Librandi, Apr 29 2014 *)

{a(n) = local(A); if( n<1, n==0, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (-4*k + 9) * A[k-1] + 3 * sum( j=1, k-1, A[j] * A[k-j])); A[n])} /* Michael Somos, Jul 23 2011 */

a(n) = 4*3^(n-2); n >= 2; a(0) = 1; a(1) = 1.
G.f.: (1-x)^2/(1-3*x).
G.f.: 1/(1-sum(j>=1, (2*j-1)*x^j )). - Joerg Arndt, Jul 06 2011
a(n) = 3*a(n-1)+(-1)^n*C(2, 2-n).
a(n) = A003946(n-1), n>0. - R. J. Mathar, Oct 13 2008
a(n) = (-4*n + 9) * a(n-1) + 3 * Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - Michael Somos, Jul 23 2011
a(n) = Sum_{k, 0<=k<=n} A201780(n,k). - Philippe Deléham, Dec 05 2011

New name from Joerg Arndt, Jul 06 2011

A055842 Expansion of (1-x)^2/(1-5*x).

Original entry on oeis.org

1, 3, 16, 80, 400, 2000, 10000, 50000, 250000, 1250000, 6250000, 31250000, 156250000, 781250000, 3906250000, 19531250000, 97656250000, 488281250000, 2441406250000, 12207031250000, 61035156250000, 305175781250000, 1525878906250000, 7629394531250000

Offset: 0

Barry E. Williams, May 30 2000

First differences of A005054.
For n>=2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4,5} we have f(x_1)<>y_1 and f(x_2)<> y_2. - Milan Janjic, Apr 19 2007
a(n) is the number of generalized compositions of n when there are 4 *i-1 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A000351, A005054.

Concatenation([1,3], List([2..30], n-> 16*5^(n-2) )); # G. C. Greubel, Jan 21 2020

[1,3] cat [16*5^(n-2): n in [2..30]]; // G. C. Greubel, Jan 21 2020

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1-x)^2/(1-5*x))); // Marius A. Burtea, Jan 21 2020

seq( `if`(n<2, 2*n+1, 16*5^(n-2)), n=0..30); # G. C. Greubel, Jan 21 2020

Join[{1,3},16 5^(Range[2,30]-2)] (* Harvey P. Dale, Apr 03 2013 *)

Vec((1-x)^2/(1-5*x) + O(x^30)) \\ Altug Alkan, Mar 13 2016

[1,3]+[16*5^(n-2) for n in (2..30)] # G. C. Greubel, Jan 21 2020

a(n) = 16*5^(n-2), a(0)=1, a(1)=3.
a(n) = 5*a(n-1) + (-1)^n*binomial(2,2-n).
G.f.: (1-x)^2/(1-5*x).
a(n) = Sum_{k=0..n} A201780(n,k)*3^k. - Philippe Deléham, Dec 05 2011
E.g.f.: (9 - 5*x + 16*exp(x))/25. - G. C. Greubel, Jan 21 2020

A055841 Number of compositions of n into 3*j-1 kinds of j's for all j >= 1.

Original entry on oeis.org

1, 2, 9, 36, 144, 576, 2304, 9216, 36864, 147456, 589824, 2359296, 9437184, 37748736, 150994944, 603979776, 2415919104, 9663676416, 38654705664, 154618822656, 618475290624, 2473901162496, 9895604649984, 39582418599936, 158329674399744, 633318697598976, 2533274790395904, 10133099161583616, 40532396646334464, 162129586585337856, 648518346341351424

Offset: 0

Barry E. Williams, May 30 2000

First differences of A002001.
For n >= 2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4} we have f(x_1) <> y_1 and f(x_2) <> y_2. - Milan Janjic, Apr 19 2007
Convolved with [1, 2, 3, ...] = powers of 4: [1, 4, 16, 64, ...]. - Gary W. Adamson, Jun 04 2009
a(n) is the number of generalized compositions of n when there are 3 *i-1 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Daniel Birmajer, Juan B. Gil, Michael D. Weiner, (an + b)-color compositions, arXiv:1707.07798 [math.CO], 2017.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A000302 and A002001.

Essentially the same as A002063.

Join[{1, 2}, 9*4^Range[0, 30]] (* Jean-François Alcover, Jul 21 2018 *)

a(n) = 9*4^(n-2), a(0)=1, a(1)=2.
a(0)=1, a(1)=2, a(3)=9, a(n+1)=4*a(n) for n >= 3.
G.f.: (1-x)^2/(1-4*x).
G.f.: 1/(1 - Sum_{j>=1} (3*j-1)*x^j). - Joerg Arndt, Jul 06 2011
a(n) = 4*a(n-1) + (-1)^n*C(2,2-n).
a(n) = Sum_{k=0..n} A201780(n,k)*2^k. - Philippe Deléham, Dec 05 2011

New name from Joerg Arndt, Jul 06 2011

A055270 a(n) = 7a(n-1) + (-1)^n binomial(2,2-n) with a(-1)=0.

Original entry on oeis.org

1, 5, 36, 252, 1764, 12348, 86436, 605052, 4235364, 29647548, 207532836, 1452729852, 10169108964, 71183762748, 498286339236, 3488004374652, 24416030622564, 170912214357948, 1196385500505636, 8374698503539452, 58622889524776164, 410360226673433148, 2872521586714032036

Offset: 0

Barry E. Williams, May 10 2000

For n >= 2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5,6,7} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4,5,6,7} we have f(x_1) <> y_1 and f(x_2) <> y_2. - Milan Janjic, Apr 19 2007

a(n) is the number of generalized compositions of n when there are 6*i-1 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 122-125, 194-196.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (7).

Cf. A055272 (first differences of 7^n (A000420)).

[1,5] cat [36*7^(n-2): n in [2..30]]; // G. C. Greubel, Mar 16 2020

A055270:= n-> `if`(n<2, 4*n+1, 36*7^(n-2)); seq(A055270(n), n=0..30); # G. C. Greubel, Mar 16 2020

Join[{1,5},NestList[7#&,36,20]] (* Harvey P. Dale, Sep 04 2017 *)

[1,5]+[36*7^(n-2) for n in (2..30)] # G. C. Greubel, Mar 16 2020

a(n) = 6^2 * 7^(n-2), n >= 2 with a(0)=1, a(1)=5.
G.f.: (1-x)^2/(1-7*x).
a(n) = Sum_{k=0..n} A201780(n,k)*5^k. - Philippe Deléham, Dec 05 2011
E.g.f.: (13 - 7*x + 36*exp(7*x))/49. - G. C. Greubel, Mar 16 2020

Terms a(20) onward added by G. C. Greubel, Mar 16 2020

A201509 Irregular triangle read by rows: T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = 0, T(n,0) = T(1,1) = 1 and T(n,k) = 0 if k < 0 or if n < k.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 1, 8, 12, 4, 16, 28, 13, 1, 32, 64, 38, 6, 64, 144, 104, 25, 1, 128, 320, 272, 88, 8, 256, 704, 688, 280, 41, 1, 512, 1536, 1696, 832, 170, 10, 1024, 3328, 4096, 2352, 620, 61, 1, 2048, 7168

Offset: 0

Paul Curtz, Dec 02 2011

This is the pseudo-triangle whose successive lines are of the type T(n,0), T(n,1)+T(n-1,0), T(n,2)+T(n-1,1), ... T(n,k)+T(n-1,k-1), without 0's, with T=A201701. [e-mail, Philippe Deléham, Dec 04 2011]

			Triangle starts:
    1   1
    2   2
    4   5   1
    8  12   4
   16  28  13  1
   32  64  38  6
   64 144 104 25 1
  128 320 272 88 8
  ...
Triangle begins (full version):
    0
    1,   1
    2,   2,   0
    4,   5,   1,  0
    8,  12,   4,  0, 0
   16,  28,  13,  1, 0, 0
   32,  64,  38,  6, 0, 0, 0
   64, 144, 104, 25, 1, 0, 0, 0
  128, 320, 272, 88, 8, 0, 0, 0, 0

Cf. A052542 (row sums).
Cf. A039991, A034867.
Cf. A201780, A263633.

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = 0, T(n,0) = T(1,1) = 1 and T(n,k) = 0 if k < 0 or if n < k. - Philippe Deléham, Dec 05 2011
The n-th row polynomial appears to equal Sum_{k = 1..floor((n+1)/2)} binomial(n,2*k-1)*(1+t)^k. Cf. A034867. - Peter Bala, Sep 10 2012
Aside from the first two rows below, the signed coefficients appear in the expansion (b*x - 1)^2 / (a*b*x^2 - 2a*x + 1) = 1 + (2 a - 2 b)x + (4 a^2 - 5 a b + b^2)x^2 + (8 a^3 - 12 a^2b + 4 ab^2)x^3 + ..., the reciprocal of the derivative of x*(1-a*x) / (1-b*x). This is related to A263633 via the expansion (a*b*x^2 - 2a*x + 1) / (b*x - 1)^2  = 1 + (b - a) (2x + 3b x^2 + 4b^2 x^3 + ...). See also A201780. - Tom Copeland, Oct 30 2023

Edited and new name using Philippe Deléham's formula, Joerg Arndt, Dec 13 2023

A055995 a(n) = 64*9^(n-2), a(0)=1, a(1)=7.

Original entry on oeis.org

1, 7, 64, 576, 5184, 46656, 419904, 3779136, 34012224, 306110016, 2754990144, 24794911296, 223154201664, 2008387814976, 18075490334784, 162679413013056, 1464114717117504, 13177032454057536, 118593292086517824

Offset: 0

Barry E. Williams, Jun 04 2000

For n>=2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5,6,7,8,9} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4,5,6,7,8,9} we have f(x_1)<>y_1 and f(x_2)<> y_2. - Milan Janjic, Apr 19 2007

a(n) is the number of generalized compositions of n when there are 8*i-1 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (9)

Second differences of 9^n (A001019). Cf. A055275.

a(n) = 9a(n-1) + ((-1)^n)*C(2, 2-n).
G.f.: (1-x)^2/(1-9x).
a(n) = Sum_{k, 0<=k<=n} A201780(n,k)*7^k. - Philippe Deléham, Dec 05 2011

A055996 a(n) = 81*10^(n-2), a(0)=1, a(1)=8.

Original entry on oeis.org

1, 8, 81, 810, 8100, 81000, 810000, 8100000, 81000000, 810000000, 8100000000, 81000000000, 810000000000, 8100000000000, 81000000000000, 810000000000000, 8100000000000000, 81000000000000000, 810000000000000000

Offset: 0

Barry E. Williams, Jun 04 2000

For n>=2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5,6,7,8,9,10} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4,5,6,7,8,9,10} we have f(x_1)<>y_1 and f(x_2)<> y_2. - Milan Janjic, Apr 19 2007

a(n) is the number of generalized compositions of n when there are 9*i-1 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (10)

Second differences of 10^n (A011557). Cf. A052268.

Join[{1,8},NestList[10#&,81,20]] (* Harvey P. Dale, Nov 20 2015 *)

a(n)=10a(n-1)+[(-1)^n]*C(2, 2-n). G.f.(x)=(1-x)^2/(1-10x).

a(n) = Sum_{k, 0<=k<=n} A201780(n,k)*8^k. - Philippe Deléham, Dec 05 2011

A056002 a(n) = (10^2)*11^(n-2); a(0)=1, a(1)=9.

Original entry on oeis.org

1, 9, 100, 1100, 12100, 133100, 1464100, 16105100, 177156100, 1948717100, 21435888100, 235794769100, 2593742460100, 28531167061100, 313842837672100, 3452271214393100, 37974983358324100, 417724816941565100

Offset: 0

Barry E. Williams, Jun 18 2000

For n>=2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5,6,7,8,9,10,11} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4,5,6,7,8,9,10,11} we have f(x_1)<>y_1 and f(x_2)<> y_2. - Milan Janjic, Apr 19 2007

a(n) is the number of generalized compositions of n when there are 10*i-1 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A001020.

Join[{1,9},100*11^Range[0,20]] (* or *) Join[{1,9},NestList[11#&,100,20]] (* Harvey P. Dale, May 24 2012 *)

a(n)=11a(n-1)+[(-1)^n]*C(2, 2-n). G.f.(x)=(1-x)^2/(1-11x).

a(n) = Sum_{k, 0<=k<=n} A201780(n,k)*9^k. - Philippe Deléham, Dec 05 2011

More terms from James Sellers, Jul 04 2000

A056116 a(n) = 121*12^(n-2), a(0)=1, a(1)=10.

Original entry on oeis.org

1, 10, 121, 1452, 17424, 209088, 2509056, 30108672, 361304064, 4335648768, 52027785216, 624333422592, 7492001071104, 89904012853248, 1078848154238976, 12946177850867712, 155354134210412544

Offset: 0

Barry E. Williams, Jul 04 2000

For n >= 2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5,6,7,8,9,10,11,12} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4,5,6,7,8,9,10,11,12} we have f(x_1)<>y_1 and f(x_2)<> y_2. - Milan Janjic, Apr 19 2007

a(n) is the number of generalized compositions of n when there are 11*i-1 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (12).

Cf. A055996, A056002.

concatenation([1,10], List([2..20], n-> 121*12^(n-2) )); # G. C. Greubel, Jan 18 2020

[1,10] cat [121*12^(n-2): n in [2..20]]; // G. C. Greubel, Jan 18 2020

1,10, seq( 121*12^(n-2), n=2..20); # G. C. Greubel, Jan 18 2020

LinearRecurrence[{12},{1,10,121},20] (* Harvey P. Dale, Oct 20 2015 *)

concat([1, 10], vector(20, n, 121*12^(n-1) )) \\ G. C. Greubel, Jan 18 2020

[1,10]+[121*12^(n-2) for n in (2..20)] # G. C. Greubel, Jan 18 2020

a(n) = 12*a(n-1) + (-1)^n*C(2, 2-n).
G.f.: (1-x)^2/(1-12*x).
a(n) = Sum_{k=0..n} A201780(n,k)*10^k. - Philippe Deléham, Dec 05 2011
E.g.f.: (23 - 12*x + 121*exp(12*x))/144. - G. C. Greubel, Jan 18 2020

More terms from James Sellers, Jul 04 2000

A055846 a(n) = 25*6^(n-2), with a(0)=1 and a(1)=4.

Original entry on oeis.org

1, 4, 25, 150, 900, 5400, 32400, 194400, 1166400, 6998400, 41990400, 251942400, 1511654400, 9069926400, 54419558400, 326517350400, 1959104102400, 11754624614400, 70527747686400, 423166486118400, 2538998916710400

Offset: 0

Barry E. Williams, Jun 03 2000

For n>=2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5,6} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4,5,6} we have f(x_1)<>y_1 and f(x_2)<> y_2. - Milan Janjic, Apr 19 2007

a(n) is the number of generalized compositions of n when there are 5*i-1 different types of i, (i=1,2,...). - Milan Janjic, Aug 26 2010

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for linear recurrences with constant coefficients, signature (6)

First differences of A052934. Cf. A000400.

LinearRecurrence[{6},{1,4,25},30] (* Harvey P. Dale, May 25 2023 *)

a(n) = 25*6^(n-2), a(0)=1, a(1)=4. a(n) = 6a(n-1) + ((-1)^n)*binomial(2, 2-n); g.f.(x)=(1-x)^2/(1-6x).

a(n) = Sum_{k, 0<=k<=n} A201780(n,k)*4^k. - Philippe Deléham, Dec 05 2011

More terms from James Sellers, Jun 05 2000

cp's OEIS Frontend

A052156 Number of compositions of n into 2*j-1 kinds of j's for all j>=1.

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A055842 Expansion of (1-x)^2/(1-5*x).

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A055841 Number of compositions of n into 3*j-1 kinds of j's for all j >= 1.

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A055270 a(n) = 7*a(n-1) + (-1)^n * binomial(2,2-n) with a(-1)=0.

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A201509 Irregular triangle read by rows: T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = 0, T(n,0) = T(1,1) = 1 and T(n,k) = 0 if k < 0 or if n < k.

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A055995 a(n) = 64*9^(n-2), a(0)=1, a(1)=7.

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A055996 a(n) = 81*10^(n-2), a(0)=1, a(1)=8.

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A055270 a(n) = 7a(n-1) + (-1)^n binomial(2,2-n) with a(-1)=0.