A003945 -id:A003945 - cp's OEIS frontend

A003953 Expansion of g.f.: (1+x)/(1-10*x).

1, 11, 110, 1100, 11000, 110000, 1100000, 11000000, 110000000, 1100000000, 11000000000, 110000000000, 1100000000000, 11000000000000, 110000000000000, 1100000000000000, 11000000000000000

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 11.
a(n) is sequence A003945(n-1) written in base 2: a(0)=1, a(n) for n >= 1: 2 times 1, (n-1) times 0. a(n) is also A007283(n-1) and A042950(n) for n >= 1 written in base 2. a(n) is also A098011(n+3) and A101229(n+10) for n >= 1 written in base 2. a(n) is also abs(A110164(n+1)) for n >= 1 written in base 2. - Jaroslav Krizek, Aug 17 2009
a(n) equals the numbers of words of length n on alphabet {0,1,...,10} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, Jun 02 2017]

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 312
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (10).
Index entries for sequences related to trees

Cf. A003945, A003952.

k:=11;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

k:=11; [1] cat [k*(k-1)^(n-1): n in [1..25]]; // G. C. Greubel, Sep 24 2019

k:=11; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # modified by G. C. Greubel, Sep 24 2019

Join[{1}, 11*10^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
Join[{1},NestList[10#&,11,20]] (* Harvey P. Dale, Jul 04 2025 *)

a(n)=11*10^n\10 \\ Charles R Greathouse IV, Aug 14 2015

k=11; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Sep 24 2019

a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 9. - Philippe Deléham, Jul 10 2005
G.f.: (1+x)/(1-10*x). - Paul Barry, Mar 22 2006
a(0) = 1, a(n) = 10^n + 10^(n-1) = 11*10^(n-1) for n >= 1. - Jaroslav Krizek, Aug 17 2009
E.g.f.: (11*exp(10*x) - 1)/10. - G. C. Greubel, Sep 24 2019

Edited by N. J. A. Sloane, Dec 04 2009

A003951 Expansion of g.f.: (1+x)/(1-8*x).

Original entry on oeis.org

1, 9, 72, 576, 4608, 36864, 294912, 2359296, 18874368, 150994944, 1207959552, 9663676416, 77309411328, 618475290624, 4947802324992, 39582418599936, 316659348799488, 2533274790395904, 20266198323167232, 162129586585337856, 1297036692682702848

Offset: 0

N. J. A. Sloane

Coordination sequence for infinite tree with valency 9.
Binomial transform is {1, 10, 91, 820, 7381, ...}, see A002452. - Philippe Deléham, Jul 22 2005
a(n) equals the number of words of length n on alphabet {0,1,...,8} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015 [Corrected by David Nacin, May 31 2017]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 310
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (8).
Index entries for sequences related to trees

Cf. A003945.

k:=9;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

[1] cat [9*8^(n-1): n in [1..25]]; // Vincenzo Librandi, Dec 11 2012

k:=9; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # modified by G. C. Greubel, Sep 24 2019

Join[{1}, 9*8^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
CoefficientList[Series[(1+x)/(1-8*x), {x, 0, 25}], x] (* Vincenzo Librandi, Dec 10 2012 *)

a(n)=if(n,9*8^n/8,1) \\ Charles R Greathouse IV, Mar 22 2016

k=9; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Sep 24 2019

a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 7. - Philippe Deléham, Jul 10 2005
a(0) = 1; for n>0, a(n) = 9*8^(n-1). - Vincenzo Librandi, Nov 18 2010
a(0) = 1, a(1) = 9, a(n) = 8*a(n-1). - Vincenzo Librandi, Dec 10 2012
E.g.f.: (9*exp(8*x) -1)/8. - G. C. Greubel, Sep 24 2019

Edited by N. J. A. Sloane, Dec 04 2009

A029653 Numbers in (2,1)-Pascal triangle (by row).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 7, 9, 5, 1, 2, 9, 16, 14, 6, 1, 2, 11, 25, 30, 20, 7, 1, 2, 13, 36, 55, 50, 27, 8, 1, 2, 15, 49, 91, 105, 77, 35, 9, 1, 2, 17, 64, 140, 196, 182, 112, 44, 10, 1, 2, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 2, 21, 100, 285

Offset: 0

Mohammad K. Azarian

Reverse of A029635. Row sums are A003945. Diagonal sums are Fibonacci(n+2) = Sum_{k=0..floor(n/2)} (2n-3k)*C(n-k,n-2k)/(n-k). - Paul Barry, Jan 30 2005
Riordan array ((1+x)/(1-x), x/(1-x)). The signed triangle (-1)^(n-k)T(n,k) or ((1-x)/(1+x), x/(1+x)) is the inverse of A055248. Row sums are A003945. Diagonal sums are F(n+2). - Paul Barry, Feb 03 2005
Row sums = A003945: (1, 3, 6, 12, 24, 48, 96, ...) = (1, 3, 7, 15, 31, 63, 127, ...) - (0, 0, 1, 3, 7, 15, 31, ...); where (1, 3, 7, 15, ...) = A000225. - Gary W. Adamson, Apr 22 2007
Triangle T(n,k), read by rows, given by (2,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 17 2011
A029653 is jointly generated with A208510 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+x*v(n-1)x and v(n,x)=u(n-1,x)+x*v(n-1,x)+1. See the Mathematica section. - Clark Kimberling, Feb 28 2012
For a closed-form formula for arbitrary left and right borders of Pascal like triangle, see A228196. - Boris Putievskiy, Aug 18 2013
For a closed-form formula for generalized Pascal's triangle, see A228576. - Boris Putievskiy, Sep 04 2013
The n-th row polynomial is (2 + x)*(1 + x)^(n-1) for n >= 1. More generally, the n-th row polynomial of the Riordan array ( (1-a*x)/(1-b*x), x/(1-b*x) ) is (b - a + x)*(b + x)^(n-1) for n >= 1. - Peter Bala, Feb 25 2018

			The triangle T(n,k) begins:
n\k 0  1  2   3   4   5   6   7  8  9 10 ...
0:  1
1:  2  1
2:  2  3  1
3:  2  5  4   1
4:  2  7  9   5   1
5:  2  9 16  14   6   1
6:  2 11 25  30  20   7   1
7:  2 13 36  55  50  27   8   1
8:  2 15 49  91 105  77  35   9  1
9:  2 17 64 140 196 182 112  44 10  1
10: 2 19 81 204 336 378 294 156 54 11  1
... Reformatted. - _Wolfdieter Lang_, Jan 09 2015
With the array M(k) as defined in the Formula section, the infinite product M(0)*M(1)*M(2)*... begins
/1        \/1         \/1        \      /1        \
|2 1      ||0 1       ||0 1      |      |2 1      |
|2 1 1    ||0 2 1     ||0 0 1    |... = |2 3 1    |
|2 1 1 1  ||0 2 1 1   ||0 0 2 1  |      |2 5 4 1  |
|2 1 1 1 1||0 2 1 1 1 ||0 0 2 1 1|      |2 7 9 5 1|
|...      ||...       ||...      |      |...      |
- _Peter Bala_, Dec 27 2014

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.
Hacene Belbachir and Athmane Benmezai, Expansion of Fibonacci and Lucas Polynomials: An Answer to Prodinger's Question, Journal of Integer Sequences, Vol. 15 (2012), #12.7.6.
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 39.
H. Hosoya, Pascal's triangle, non-adjacent numbers and D-dimensional atomic orbitals, J. Math. Chemistry, vol. 23, 1998, 169-178.
M. Janjic and B. Petkovic, A Counting Function, arXiv preprint arXiv:1301.4550 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 13 2013
M. Janjic and B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 8.
Mark C. Wilson, Asymptotics for generalized Riordan arrays. International Conference on Analysis of Algorithms DMTCS proc. AD. Vol. 323. 2005.

(d, 1) Pascal triangles: A007318(d=1), A093560(3), A093561(4), A093562(5), A093563(6), A093564(7), A093565(8), A093644(9), A093645(10).
Cf. A003945, A208510, A228196, A228576.
Cf. A078812, A106195.

a029653 n k = a029653_tabl !! n !! k
a029653_row n = a029653_tabl !! n
a029653_tabl = [1] : iterate
               (\xs -> zipWith (+) ([0] ++ xs) (xs ++ [0])) [2, 1]
-- Reinhard Zumkeller, Dec 16 2013

A029653 :=  proc(n,k)
if n = 0 then
  1;
else
  binomial(n-1, k)+binomial(n, k)
fi
end proc: # R. J. Mathar, Jun 30 2013

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A208510 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A029653 *)
(* Clark Kimberling, Feb 28 2012 *)

from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1
def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017

from math import comb, isqrt
def A029653(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),a:=n-comb(r+1,2))*((r<<1)-a)//r if n else 1 # Chai Wah Wu, Nov 12 2024

T(n, k) = C(n-2, k-1) + C(n-2, k) + C(n-1, k-1) + C(n-1, k) except for n=0.
G.f.: (1 + x + y + xy)/(1 - y - xy). - Ralf Stephan, May 17 2004
T(n, k) = (2n-k)*binomial(n, n-k)/n, n, k > 0. - Paul Barry, Jan 30 2005
Sum_{k=0..n} T(n, k)*x^k gives A003945-A003954 for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. - Philippe Deléham, Jul 10 2005
T(n, k) = C(n-1, k) + C(n, k). - Philippe Deléham, Jul 10 2005
Equals A097806 * A007318, i.e., the pairwise operator * Pascal's Triangle as infinite lower triangular matrices. - Gary W. Adamson, Apr 22 2007
From Peter Bala, Dec 27 2014: (Start)
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(2 + 5*x + 4*x^2/2! + x^3/3!) = 2 + 7*x + 16*x^2/2! + 30*x^3/3! + 50*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ).
Let M denote the lower unit triangular array with 1's on the main diagonal and 1's everywhere else below the main diagonal except for the first column which consists of the sequence [1,2,2,2,...]. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)

More terms from James Sellers

A170758 Expansion of g.f.: (1+x)/(1-38*x).

Original entry on oeis.org

1, 39, 1482, 56316, 2140008, 81320304, 3090171552, 117426518976, 4462207721088, 169563893401344, 6443427949251072, 244850262071540736, 9304309958718547968, 353563778431304822784, 13435423580389583265792, 510546096054804164100096

Offset: 0

N. J. A. Sloane, Dec 04 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (38).

Cf. A003945.

k:=39;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Oct 09 2019

[1] cat [39*38^(n-1): n in [1..20]]; // Vincenzo Librandi, Apr 28 2014

k:=39; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # G. C. Greubel, Oct 09 2019

CoefficientList[Series[(1+x)/(1-38x), {x, 0, 20}], x] (* Vincenzo Librandi, Apr 28 2014 *)
With[{k = 39}, Table[If[n==0, 1, k*(k-1)^(n-1)], {n, 0, 25}]] (* G. C. Greubel, Oct 09 2019 *)
Join[{1},NestList[38#&,39,20]] (* Harvey P. Dale, Aug 07 2025 *)

vector(26, n, k=39; if(n==1, 1, k*(k-1)^(n-2))) \\ G. C. Greubel, Oct 09 2019

k=39; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Oct 09 2019

a(n) = Sum_{k=0..n} A097805(n,k)*(-1)^(n-k)*39^k. - Philippe Deléham, Dec 04 2009
a(0)=1; for n>0, a(n) = 39*38^(n-1). - Vincenzo Librandi, Dec 05 2009
E.g.f.: (39*exp(38*x) - 1)/38. - G. C. Greubel, Oct 09 2019

A170732 Expansion of g.f.: (1+x)/(1 - 12*x).

Original entry on oeis.org

1, 13, 156, 1872, 22464, 269568, 3234816, 38817792, 465813504, 5589762048, 67077144576, 804925734912, 9659108818944, 115909305827328, 1390911669927936, 16690940039135232, 200291280469622784, 2403495365635473408, 28841944387625680896, 346103332651508170752

Offset: 0

N. J. A. Sloane, Dec 05 2009

For n >= 1, a(n) equals the number of words of length n-1 on the alphabet {0,1,...,12} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015

Kenny Lau, Table of n, a(n) for n = 0..926
M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015) # 15.4.7.
Index entries for linear recurrences with constant coefficients, signature (12).

Cf. A003945, A003953, A003954.

k:=13;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

k:=13; [1] cat [k*(k-1)^(n-1): n in [1..25]]; // G. C. Greubel, Sep 24 2019

k:=13; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # G. C. Greubel, Sep 24 2019

Join[{1}, 13*12^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
Join[{1},NestList[12#&,13,20]] (* Harvey P. Dale, Nov 24 2024 *)

a(n)=if(n,13*12^(n-1),1) \\ Charles R Greathouse IV, Jul 01 2016

for i in range(1001):print(i,13*12**(i-1) if i>0 else 1) # Kenny Lau, Aug 01 2017

k=13; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Sep 24 2019

a(0)=1; for n > 0, a(n) = 13*12^(n-1). - Vincenzo Librandi, Dec 05 2009

E.g.f.: (13*exp(12*x) - 1)/12. - G. C. Greubel, Sep 24 2019

A170733 Expansion of g.f.: (1+x)/(1-13*x).

Original entry on oeis.org

1, 14, 182, 2366, 30758, 399854, 5198102, 67575326, 878479238, 11420230094, 148462991222, 1930018885886, 25090245516518, 326173191714734, 4240251492291542, 55123269399790046, 716602502197270598, 9315832528564517774, 121105822871338731062, 1574375697327403503806

Offset: 0

N. J. A. Sloane, Dec 04 2009

For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,...,13} with no two adjacent letters identical. - Milan Janjic, Jan 31 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..900
M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015) # 15.4.7.
Index entries for linear recurrences with constant coefficients, signature (13).

Cf. A003945, A003954, A170732.

k:=14;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

k:=14; [1] cat [k*(k-1)^(n-1): n in [1..25]]; // G. C. Greubel, Sep 24 2019

k:=14; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # G. C. Greubel, Sep 24 2019

Join[{1}, 14*13^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
CoefficientList[Series[(1+x)/(1-13x), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 10 2012 *)
Join[{1},NestList[13#&,14,20]] (* Harvey P. Dale, Oct 09 2017 *)

vector(26, n, k=14; if(n==1, 1, k*(k-1)^(n-2))) \\ G. C. Greubel, Sep 24 2019

k=14; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Sep 24 2019

a(n) = Sum_{k=0..n} A097805(n,k)*(-1)^(n-k)*14^k. -  Philippe Deléham, Dec 04 2009
a(0) = 1; for n>0, a(n) = 14*13^(n-1). - Vincenzo Librandi, Dec 05 2009
a(0)=1, a(1)=14, a(n) = 13*a(n-1). - Vincenzo Librandi, Dec 10 2012
E.g.f.: (14*exp(13*x) - 1)/13. - G. C. Greubel, Sep 24 2019

A170749 Expansion of g.f.: (1+x)/(1-29*x).

Original entry on oeis.org

1, 30, 870, 25230, 731670, 21218430, 615334470, 17844699630, 517496289270, 15007392388830, 435214379276070, 12621216999006030, 366015292971174870, 10614443496164071230, 307818861388758065670, 8926746980273983904430, 258875662427945533228470

Offset: 0

N. J. A. Sloane, Dec 04 2009

Kenny Lau, Table of n, a(n) for n = 0..683
Index entries for linear recurrences with constant coefficients, signature (29).

Cf. A003945, A097805.

k:=30;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 25 2019

k:=30; [1] cat [k*(k-1)^(n-1): n in [1..25]]; // G. C. Greubel, Sep 25 2019

k:=30; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # G. C. Greubel, Sep 25 2019

With[{k=30}, Table[If[n==0,1, k*(k-1)^(n-1)], {n,0,25}]] (* G. C. Greubel, Sep 25 2019 *)
Join[{1},NestList[29#&,30,20]] (* Harvey P. Dale, Aug 27 2020 *)

vector(26, n, k=30; if(n==1, 1, k*(k-1)^(n-2))) \\ G. C. Greubel, Sep 25 2019

for i in range(31):print(i,30*29**(i-1) if i>0 else 1) # Kenny Lau, Aug 03 2017

k=30; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Sep 25 2019

a(n) = Sum_{k=0..n} A097805(n,k)*(-1)^(n-k)*30^k. - Philippe Deléham, Dec 04 2009
a(0) = 1; for n > 0, a(n) = 30*29^(n-1). - Vincenzo Librandi, Dec 05 2009
E.g.f.: (30*exp(29*x) -1)/29. - G. C. Greubel, Sep 25 2019

A170762 Expansion of g.f.: (1+x)/(1-42*x).

Original entry on oeis.org

1, 43, 1806, 75852, 3185784, 133802928, 5619722976, 236028364992, 9913191329664, 416354035845888, 17486869505527296, 734448519232146432, 30846837807750150144, 1295567187925506306048, 54413821892871264854016, 2285380519500593123868672

Offset: 0

N. J. A. Sloane, Dec 04 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries for linear recurrences with constant coefficients, signature (42).

Cf. A003945.

k:=43;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Oct 10 2019

k:=43; [1] cat [k*(k-1)^(n-1): n in [1..25]]; // G. C. Greubel, Oct 10 2019

k:=43; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # G. C. Greubel, Oct 10 2019

CoefficientList[Series[(1+x)/(1-42x),{x,0,30}],x] (* or *) Join[{1}, NestList[42#&,43,30]] (* Harvey P. Dale, Mar 26 2012 *)
With[{k = 43}, Table[If[n==0, 1, k*(k-1)^(n-1)], {n, 0, 25}]] (* G. C. Greubel, Oct 10 2019 *)

a(n)=if(n,43*42^(n-1),1) \\ Charles R Greathouse IV, Mar 22 2016

k=43; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Oct 10 2019

a(n) = Sum_{k=0..n} A097805(n,k)*(-1)^(n-k)*43^k. - Philippe Deléham, Dec 04 2009
a(0) = 1; for n>0, a(n) = 43*42^(n-1). - Vincenzo Librandi, Dec 05 2009
a(0)=1, a(1)=43, a(n)=42*a(n-1). - Harvey P. Dale, Mar 26 2012
E.g.f.: (43*exp(42*x) - 1)/42. - G. C. Greubel, Oct 10 2019

A170769 Expansion of g.f.: (1+x)/(1-49*x).

Original entry on oeis.org

1, 50, 2450, 120050, 5882450, 288240050, 14123762450, 692064360050, 33911153642450, 1661646528480050, 81420679895522450, 3989613314880600050, 195491052429149402450, 9579061569028320720050, 469374016882387715282450, 22999326827236998048840050

Offset: 0

N. J. A. Sloane, Dec 04 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..600
Index entries for linear recurrences with constant coefficients, signature (49).

Cf. A003945.

k:=50;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Oct 10 2019

k:=50; [1] cat [k*(k-1)^(n-1): n in [1..25]]; // G. C. Greubel, Oct 10 2019

k:=50; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # G. C. Greubel, Oct 10 2019

CoefficientList[Series[(1+x)/(1-49*x), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 09 2012 *)
With[{k = 50}, Table[If[n==0, 1, k*(k-1)^(n-1)], {n, 0, 25}]] (* G. C. Greubel, Oct 10 2019 *)

A170769(n):=if n=0 then 1 else 50*49^(n-1)$
makelist(A170769(n),n,0,30); /* Martin Ettl, Nov 06 2012 */

vector(26, n, k=50; if(n==1, 1, k*(k-1)^(n-2))) \\ G. C. Greubel, Oct 10 2019

k=50; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Oct 10 2019

a(n) = Sum_{k=0..n} A097805(n,k)*(-1)^(n-k)*50^k. - Philippe Deléham, Dec 04 2009
a(0) = 1; for n>0, a(n) = 50*49^(n-1). - Vincenzo Librandi, Dec 05 2009
E.g.f.: (50*exp(49*x) - 1)/49. - G. C. Greubel, Oct 11 2019

A170734 Expansion of g.f.: (1+x)/(1-14*x).

Original entry on oeis.org

1, 15, 210, 2940, 41160, 576240, 8067360, 112943040, 1581202560, 22136835840, 309915701760, 4338819824640, 60743477544960, 850408685629440, 11905721598812160, 166680102383370240, 2333521433367183360, 32669300067140567040, 457370200939967938560, 6403182813159551139840

Offset: 0

N. J. A. Sloane, Dec 04 2009

For n>=1, a(n) equals the numbers of words of length n-1 on alphabet {0,1,...,14} with no two adjacent letters identical. -Milan Janjic, Jan 31 2015

Vincenzo Librandi, Table of n, a(n) for n = 0..800
M. Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, J. Int. Seq. 18 (2015) # 15.4.7.
Index entries for linear recurrences with constant coefficients, signature (14).

Cf. A003945, A003954, A097805, A170733.

k:=15;; Concatenation([1], List([1..25], n-> k*(k-1)^(n-1) )); # G. C. Greubel, Sep 24 2019

k:=15; [1] cat [k*(k-1)^(n-1): n in [1..25]]; // G. C. Greubel, Sep 24 2019

k:=15; seq(`if`(n=0, 1, k*(k-1)^(n-1)), n = 0..25); # G. C. Greubel, Sep 24 2019

Join[{1}, 15*14^Range[0, 25]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
CoefficientList[Series[(1+x)/(1-14x), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 10 2012 *)

vector(26, n, k=15; if(n==1, 1, k*(k-1)^(n-2))) \\ G. C. Greubel, Sep 24 2019

k=15; [1]+[k*(k-1)^(n-1) for n in (1..25)] # G. C. Greubel, Sep 24 2019

a(n) = Sum_{k=0..n} A097805(n,k)*(-1)^(n-k)*15^k. - Philippe Deléham, Dec 04 2009
a(0) = 1; for n>0, a(n) = 15*14^(n-1). - Vincenzo Librandi, Dec 05 2009
a(0) = 1, a(1) = 15, a(n) = 14*a(n-1). - Vincenzo Librandi, Dec 10 2012
E.g.f.: (15*exp(14*x) -1)/14. - G. C. Greubel, Sep 24 2019

cp's OEIS Frontend

A003953 Expansion of g.f.: (1+x)/(1-10*x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A003951 Expansion of g.f.: (1+x)/(1-8*x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

A029653 Numbers in (2,1)-Pascal triangle (by row).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Python

Formula

Extensions

A170758 Expansion of g.f.: (1+x)/(1-38*x).

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A170732 Expansion of g.f.: (1+x)/(1 - 12*x).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Python

Sage

Formula