A016127 -id:A016127 - cp's OEIS frontend

A096035 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^4-M)/3, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

1, 5, 2, 21, 15, 3, 85, 84, 30, 4, 341, 425, 210, 50, 5, 1365, 2046, 1275, 420, 75, 6, 5461, 9555, 7161, 2975, 735, 105, 7, 21845, 43688, 38220, 19096, 5950, 1176, 140, 8, 87381, 196605, 196596, 114660, 42966, 10710, 1764, 180, 9, 349525, 873810, 983025

Offset: 1

Gary W. Adamson, Jun 17 2004

			Triangle begins:
1;
5,       2;
21,     15,    3;
85,     84,   30,   4;
341,   425,  210,  50,  5;
1365, 2046, 1275, 420, 75,  6;

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A007318. First column gives A002450. Row sums give A016127.

P:= proc(n) option remember; local M; M:= Matrix(n, (i, j)-> binomial(i-1, j-1)); (M^4-M)/3 end: T:= (n, k)-> P(n+1)[n+1, k]: seq(seq(T(n, k), k=1..n), n=1..11);  # Alois P. Heinz, Oct 07 2009

rows=11; M[n_] := M[n] = Array[Binomial, {n, n}, {0, 0}]; P[n_] := (MatrixPower[ M[n], 4] - M[n])/3; Table[P[rows+1][[n+1, 1 ;; n]], {n, 1, rows}] // Flatten (* Jean-François Alcover, Mar 11 2015 *)

Edited and more terms from Alois P. Heinz, Oct 07 2009

A327317 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = 2^(n-1) ((x+r)^n - (x+s)^n)/(r - s), where r = 2 and s = 1/2.

Original entry on oeis.org

1, 5, 4, 21, 30, 12, 85, 168, 120, 32, 341, 850, 840, 400, 80, 1365, 4092, 5100, 3360, 1200, 192, 5461, 19110, 28644, 23800, 11760, 3360, 448, 21845, 87376, 152880, 152768, 95200, 37632, 8960, 1024, 87381, 393210, 786384, 917280, 687456, 342720, 112896

Offset: 1

Clark Kimberling, Nov 03 2019

p(x,n) is a strong divisibility sequence of polynomials. That is, gcd(p(x,h),p(x,k)) = p(x,gcd(h,k)). If x is an integer, then p(x,n) is a strong divisibility sequence of integers.

			First six rows:
     1;
     5,    4;
    21,   30,   12;
    85,  168,  120,   32;
   341,  850,  840,  400,   80;
  1365, 4092, 5100, 3360, 1200, 192;
The first six polynomials, not factored:
1, 5 + 4 x, 21 + 30 x + 12 x^2, 85 + 168 x + 120 x^2 + 32 x^3, 341 + 850 x + 840 x^2 + 400 x^3 + 80 x^4, 1365 + 4092 x + 5100 x^2 + 3360 x^3 + 1200 x^4 + 192 x^5.
The first six polynomials, factored:
1, 5 + 4 x, 3 (7 + 10 x + 4 x^2), (5 + 4 x) (17 + 20 x + 8 x^2), 341 + 850 x + 840 x^2 + 400 x^3 + 80 x^4, 3 (5 + 4 x) (7 + 10 x + 4 x^2) (13 + 10 x + 4 x^2).

Cf. A327316, A002450 (x=0), A016137 (x=1), A001045 (x = -1), A016162 (x = 2), A016181 (x = 3), A016127 (x = -3), A016157 (x = 1/2).

r = 2; s = 1/2; f[x_, n_] := 2^(n - 1) ((x + r)^n - (x + s)^n)/(r - s);
Column[Table[Expand[f[x, n]], {n, 1, 5}]]
c[x_, n_] := CoefficientList[Expand[f[x, n]], x]
TableForm[Table[c[x, n], {n, 1, 10}]] (* A327317 array *)
Flatten[Table[c[x, n], {n, 1, 12}]]   (* A327317 sequence *)

A385671 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(4n+1,k) binomial(4*n-k,n-k).

Original entry on oeis.org

1, 22, 774, 30458, 1260886, 53731512, 2333065354, 102643195068, 4559878830006, 204091261040552, 9189096061165784, 415734554486178378, 18884084064916032026, 860673634902720476392, 39339618388269633525564, 1802605962076744803396888, 82777622289467318635747446

Offset: 0

Seiichi Manyama, Aug 04 2025

nonn

Cf. A016127, A385669, A385670.

Cf. A384366, A386768.

a(n) = sum(k=0, n, 2^k*3^(n-k)*binomial(4*n+1, k)*binomial(4*n-k, n-k));

a(n) = [x^n] (1+2*x)^(4*n+1)/(1-3*x)^(3*n+1).
a(n) = [x^n] 1/((1-2*x) * (1-5*x)^(3*n+1)).
a(n) = Sum_{k=0..n} 5^k * (-3)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * binomial(3*n+k,k).

A016282 Expansion of 1/((1-2x)(1-4x)(1-5*x)).

Original entry on oeis.org

1, 11, 83, 535, 3171, 17871, 97483, 520055, 2731091, 14179231, 72992283, 373347975, 1900290211, 9635660591, 48715157483, 245723238295, 1237206060531, 6220389909951, 31239388241083, 156746696495015, 785932504682051, 3938458614335311, 19727477439571083

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (11,-38,40).

Cf. A006516, A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

CoefficientList[ Series[ 1/((1 - 2x)(1 - 4x)(1 - 5x)), {x, 0, 20} ], x ]
LinearRecurrence[{11,-38,40},{1,11,83},30] (* Harvey P. Dale, Nov 29 2022 *)

Vec(1/((1-2*x)*(1-4*x)*(1-5*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[(5^n - 2^n)/3-(4^n - 2^n)/2 for n in range(2,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = (2/3)*2^n - 8*(4)^n + (25/3)*5^n. - Antonio Alberto Olivares, May 12 2012

A016295 Expansion of 1/((1-2x)(1-5x)(1-6x)).

Original entry on oeis.org

1, 13, 117, 905, 6461, 43953, 289717, 1868425, 11861421, 74423393, 462815717, 2858273145, 17556537181, 107373722833, 654414852117, 3977351721065, 24118423433741, 145982106270273, 882250466222917

Offset: 0

N. J. A. Sloane

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-52,60).

Cf. A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

LinearRecurrence[{13,-52,60},{1,13,117},20] (* Harvey P. Dale, Mar 26 2016 *)

[(6^n - 2^n)/4-(5^n - 2^n)/3 for n in range(2,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = A016129(n+1) - A016127(n+1). - Zerinvary Lajos, Jun 05 2009
a(n) = 13*a(n-1) - 52*a(n-2) + 60*a(n-3), n >= 3.
a(n) = 11*a(n-1) - 30*a(n-2) + 2^n, n >= 2. - Vincenzo Librandi, Mar 16 2011
a(n) = 7*a(n-1) - 10*a(n-2) + 6^n, n >= 2. - Vincenzo Librandi, Mar 16 2011
a(n) = 8*a(n-1) - 12*a(n-2) + 5^n, n >= 2. - Vincenzo Librandi, Mar 16 2011
a(n) = -5^(n+2)/3 + 9*6^n + 2^n/3. - R. J. Mathar, Mar 18 2011

A016633 Expansion of g.f. 1/((1-2x)(1-11x)(1-12*x)).

Original entry on oeis.org

1, 25, 447, 6989, 101759, 1417941, 19180519, 253983853, 3309800367, 42599540357, 542895780791, 6863463633117, 86197420501375, 1076563471968373, 13382900349107463, 165700329729679181, 2044564737700501583, 25152545442794015589, 308625999807796411735, 3778261997130507936445

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (25,-178,264).

Cf. A006516, A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

Cf. A016135, A016136.

[(648*12^n +2^(n+1)-5*11^(n+2))/45 : n in [0..20]]; // Vincenzo Librandi, Oct 09 2011

CoefficientList[Series[1/((1 - 2 x) (1 - 11 x) (1 - 12 x)), {x, 0, 15}], x] (* Michael De Vlieger, Jan 31 2018 *)

Vec(1/((1-2*x)*(1-11*x)*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[(12^n - 2^n)/10-(11^n - 2^n)/9 for n in range(2,18)] # Zerinvary Lajos, Jun 05 2009

From Vincenzo Librandi, Oct 09 2011: (Start)
a(n) = (648*12^n + 2^(n+1) - 5*11^(n+2))/45.
a(n) = 23*a(n-1) - 132*a(n-2) + 2^n.
a(n) = 25*a(n-1) - 178*a(n-2) + 264*a(n-3), n >= 3. (End)
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(2*x)*(648*exp(10*x) - 605*exp(9*x) + 2)/45.
a(n) = A016136(n+1) - A016135(n+1). (End)

A025992 Expansion of 1/((1-2x)(1-5x)(1-7x)(1-8*x)).

Original entry on oeis.org

1, 22, 313, 3666, 38493, 377286, 3529681, 31947322, 282198565, 2447183310, 20920905369, 176852694018, 1481626607917, 12322682753494, 101879323774177, 838170485025354, 6867569457133749, 56077266261254238

Offset: 0

N. J. A. Sloane

From Bruno Berselli, May 09 2013: (Start)
a(n) - 2*a(n-1), for n>0, gives A019928 (after 1);
a(n) - 5*a(n-1), for n>0, gives A016311 (after 1);
a(n) - 7*a(n-1), for n>0, gives A016297 (after 1);
a(n) - 8*a(n-1), for n>0, gives A016296 (after 1);
a(n) - 7*a(n-1) + 10*a(n-2), for n>1, gives A016177 (after 15);
a(n) - 9*a(n-1) + 14*a(n-2), for n>1, gives A016162 (after 13);
a(n) - 10*a(n-1) + 16*a(n-2), for n>1, gives A016161 (after 12);
a(n) - 12*a(n-1) + 35*a(n-2), for n>1, gives A016131 (after 10);
a(n) - 13*a(n-1) + 40*a(n-2), for n>1, gives A016130 (after 9);
a(n) - 15*a(n-1) + 56*a(n-2), for n>1, gives A016127 (after 7);
a(n) - 20*a(n-1) +131*a(n-2) -280*a(n-3), for n>2, gives A000079 (after 4);
a(n) - 17*a(n-1) +86*a(n-2) -112*a(n-3), for n>2, gives A000351 (after 25);
a(n) - 15*a(n-1) +66*a(n-2) -80*a(n-3), for n>2, gives A000420 (after 49);
a(n) - 14*a(n-1) +59*a(n-2) -70*a(n-3), for n>2, gives A001018 (after 64),
and naturally: a(n) - 22*a(n-1) + 171*a(n-2) - 542*a(n-3) + 560*a(n-4), for n>3, gives 0 (see Harvey P. Dale in Formula lines). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (22,-171,542,-560).

Cf. A000079, A000351, A000420, A001018, A016127, A016130, A016131, A016161, A016162, A016177, A016296, A016297, A016311, A019928.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!(1/((1-2*x)*(1-5*x)*(1-7*x)*(1-8*x)))); // Bruno Berselli, May 09 2013

CoefficientList[Series[1/((1-2x)(1-5x)(1-7x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[ {22,-171,542,-560},{1,22,313,3666},30] (* Harvey P. Dale, Jan 29 2013 *)

a(n) = n+=3; (5*8^n-9*7^n+5*5^n-2^n)/90 \\ Charles R Greathouse IV, Oct 03 2016

def A025992(n): return (5*pow(8,n+3)-9*pow(7,n+3)+pow(5,n+4)-pow(2,n+3))//90
print([A025992(n) for n in range(41)]) # G. C. Greubel, Dec 27 2024

a(0)=1, a(1)=22, a(2)=313, a(3)=3666, a(n) = 22*a(n-1) - 171*a(n-2) + 542*a(n-3) - 560*a(n-4). - Harvey P. Dale, Jan 29 2013
a(n) = (5*8^(n+3) - 9*7^(n+3) + 5^(n+4) - 2^(n+3))/90. - Yahia Kahloune, May 07 2013
E.g.f.: (1/90)*(-8*exp(2*x) + 625*exp(5*x) - 3087*exp(7*x) + 2560*exp(8*x)). - G. C. Greubel, Dec 27 2024

A016198 Expansion of g.f. 1/((1-x)(1-2x)(1-5x)).

Original entry on oeis.org

1, 8, 47, 250, 1281, 6468, 32467, 162590, 813461, 4068328, 20343687, 101722530, 508620841, 2543120588, 12715635707, 63578244070, 317891351421, 1589457019248, 7947285620527, 39736429151210, 198682147853201, 993410743460308, 4967053725690147, 24835268645227950

Offset: 0

N. J. A. Sloane, Dec 11 1999

Index entries for linear recurrences with constant coefficients, signature (8,-17,10).

Cf. A000225, A000392, A002275, A002452, A003462, A003463, A003464, A016123, A016125, A016208, A016209, A016218, A016256, A023000, A023001.

Cf. A016127.

a:=n->sum((5^(n-j)-2^(n-j))/3,j=0..n): seq(a(n), n=1..20); # Zerinvary Lajos, Jan 04 2007

Join[{a=1,b=8},Table[c=7*b-10*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011 *)

a(n) = (25*5^n - 16*2^n + 3)/12. - Bruno Berselli, Feb 09 2011
a(n) = [(5^0-2^0) + (5^1-2^1) + ... + (5^n-2^n)]/3. - r22lou(AT)cox.net, Nov 14 2005
a(0)=1, a(n) = 5*a(n-1) + 2^(n+1) - 1. - Vincenzo Librandi, Feb 07 2011
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(x)*(25*exp(4*x) - 16*exp(x) + 3)/12.
a(n) = 8*a(n-1) - 17*a(n-2) + 10*a(n-3).
a(n) = A016127(n+1) - A003463(n+2). (End)

More terms from Wesley Ivan Hurt, May 05 2014

A093953 a(n) = rightmost term in M^n * [1,1,1], where M = a 3 X 3 matrix composed of the first 3 rows of A050166 (fill in the matrix with zeros): = [1 0 0 / 1 2 0 / 1 4 5].

Original entry on oeis.org

1, 10, 63, 344, 1781, 9030, 45403, 227524, 1138641, 5695250, 28480343, 142409904, 712065901, 3560362270, 17801876883, 89009515484

Offset: 0

Gary W. Adamson, Apr 18 2004

A sequence relating to Catalan numbers.
a(n)/a(n-1) tends to 5, a Catalan number. E.g. a(6)/a(5) = 45403/9030 = 4.9948...
Generally, with M = an N X N matrix composed of rows of A050166 (along with zeros), M^n * [1,1,1...] generates terms [a, b, c, d...] such that sequences of which a,b,c,d...are members converge upon the Catalan numbers: 1, 2, 5, 14, 42, 132...
Companion (M^n)[3,2] = 4*A016127(n), (M^n)[3,3] = 5^n = A000351(n), so a(n) = a(n-1) + 4*A016127(n-1) + 5^(n-1) for n>0. - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 30 2005

			a(4) = 1781 since M^4 * [1,1,1] = [1, 31, 1781].

Index entries for linear recurrences with constant coefficients, signature (8,-17,10).

Cf. A050166, A016127, A000351.

M=[1,0,0;1,2,0;1,4,5];for(i=0,16,print1((M^i)[3,1]", ")) \\ Klasen

Or simply with M=[1, 0, 0;1, 2, 0;1, 4, 5], a(n)=(M^n)[3, 1], (adds a leading 0 to sequence). - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 30 2005

G.f.: (1+2*x)/(1-8*x+17*x^2-10*x^3). - Colin Barker, Jan 31 2012

A102752 Array read by antidiagonals: T(n, k) = ((n+2)^k-(n-1)^k)/3.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 5, 9, 5, 0, 1, 7, 21, 27, 11, 0, 1, 9, 39, 85, 81, 21, 0, 1, 11, 63, 203, 341, 243, 43, 0, 1, 13, 93, 405, 1031, 1365, 729, 85, 0, 1, 15, 129, 715, 2511, 5187, 5461, 2187, 171, 0, 1, 17, 171, 1157, 5261, 15309, 25999, 21845, 6561, 341, 0, 1

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 09 2005

Consider a 3 X 3 matrix M =
  [n, 1, 1]
  [1, n, 1]
  [1, 1, n].
The n-th row of the array contains the values of the nondiagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = nondiagonal entry + (n-1)^k.)
Table:
  n: row sequence G.f. cross references.
  0: (2^n-(-1)^n)/3 1/((1+1x)(1-2x)) A001045 (Jacobsthal sequence)
  1: (3^n-0^n)/3 1/(1-3x) A000244
  2: (4^n-1^n)/3 1/((1-1x)(1-4x)) A002450
  3: (5^n-2^n)/3 1/((1-2x)(1-5x)) A016127
  4: (6^n-3^n)/3 1/((1-3x)(1-6x)) A016137
  5: (7^n-4^n)/3 1/((1-4x)(1-7x)) A016150
  6: (8^n-5^n)/3 1/((1-5x)(1-8x)) A016162
  7: (9^n-6^n)/3 1/((1-6x)(1-9x)) A016172
  8: (10^n-7^n)/3 1/((1-7x)(1-10x)) A016181
  9: (11^n-8^n)/3 1/((1-8x)(1-11x)) A016187
  10:(12^n-9^n)/3 1/((1-9x)(1-12x)) A016191
If r(n) denotes a row sequence, r(n+1)/r(n) converges to n+2.
Columns follow polynomials with certain binomial coefficients:
  column: polynomial
  0: 0
  1: 1
  2: 2*n + 1
  3: 3*n^2+ 3*n + 3
  4: 4*n^3+ 6*n^2+ 12*n + 5
  5: 5*n^4+10*n^3+ 30*n^2+ 25*n + 11
  6: 6*n^5+15*n^4+ 60*n^3+ 75*n^2+ 66*n + 21
  7: 7*n^6+21*n^5+105*n^4+ 175*n^3+ 231*n^2+ 147*n + 43
  8: 8*n^7+28*n^6+168*n^5+ 350*n^4+ 616*n^3+ 588*n^2+344*n+ 85
  etc.
Coefficients are generated by the array T(n,k)=(2^(n-k-1)-(-1)^(n-k-1))/3*(binomial(k+(n-k-1),n-k-1)) read by antidiagonals.

			Array begins:
  0, 1, 1,  3,   5,   11, ...
  0, 1, 3,  9,  27,   81, ...
  0, 1, 5, 21,  85,  341, ...
  0, 1, 7, 39, 203, 1031, ...
  0, 1, 9, 63, 405, 2511, ...
  ...

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=0,10, for(i=0,10,print1((MM(3,k)^i)[1,2],","));print())

cp's OEIS Frontend

A096035 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^4-M)/3, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

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A327317 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = 2^(n-1) ((x+r)^n - (x+s)^n)/(r - s), where r = 2 and s = 1/2.

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A385671 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).

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A016282 Expansion of 1/((1-2*x)*(1-4*x)*(1-5*x)).

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A016295 Expansion of 1/((1-2x)(1-5x)(1-6x)).

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Sage

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A016633 Expansion of g.f. 1/((1-2*x)*(1-11*x)*(1-12*x)).

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A025992 Expansion of 1/((1-2*x)*(1-5*x)*(1-7*x)*(1-8*x)).

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A016198 Expansion of g.f. 1/((1-x)*(1-2*x)*(1-5*x)).

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A385671 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(4n+1,k) binomial(4*n-k,n-k).

A016282 Expansion of 1/((1-2x)(1-4x)(1-5*x)).

A016633 Expansion of g.f. 1/((1-2x)(1-11x)(1-12*x)).

A025992 Expansion of 1/((1-2x)(1-5x)(1-7x)(1-8*x)).

A016198 Expansion of g.f. 1/((1-x)(1-2x)(1-5x)).