A052529 -id:A052529 - cp's OEIS frontend

A127893 Riordan array (1/(1-x)^3, x/(1-x)^3).

1, 3, 1, 6, 6, 1, 10, 21, 9, 1, 15, 56, 45, 12, 1, 21, 126, 165, 78, 15, 1, 28, 252, 495, 364, 120, 18, 1, 36, 462, 1287, 1365, 680, 171, 21, 1, 45, 792, 3003, 4368, 3060, 1140, 231, 24, 1, 55, 1287, 6435, 12376, 11628, 5985, 1771, 300, 27, 1

Offset: 0

Paul Barry, Feb 04 2007

Inverse is A127894.
From Peter Bala, Jul 22 2014: (Start)
Let M denote the unsigned version of the lower unit triangular array A122432 and 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 matrix product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)

			Triangle begins
   1;
   3,    1;
   6,    6,     1;
  10,   21,     9,     1;
  15,   56,    45,    12,     1;
  21,  126,   165,    78,    15,     1;
  28,  252,   495,   364,   120,    18,     1;
  36,  462,  1287,  1365,   680,   171,    21,    1;
  45,  792,  3003,  4368,  3060,  1140,   231,   24,   1;
  55, 1287,  6435, 12376, 11628,  5985,  1771,  300,  27,  1;
  66, 2002, 12870, 31824, 38760, 26334, 10626, 2600, 378, 30, 1;
  ...
From _Peter Bala_, Jul 22 2014: (Start)
With the arrays M(k) as defined in the Comments section, the infinite product M(0*)M(1)*M(2)*... begins
  / 1         \/1         \/1       \       / 1       \
  | 3  1      ||0  1      ||0 1      |      | 3  1    |
  | 6  3 1    ||0  3 1    ||0 0 1    |... = | 6  6 1  |
  |10  6 3 1  ||0  6 3 1  ||0 0 3 1  |      |10 21 9 1|
  |15 10 6 3 1||0 10 6 3 1||0 0 6 3 1|      |...      |
  |...        ||...       ||...      |      |...      |
(End)

G. C. Greubel, Rows n = 0..99, flattened
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.

Cf. A052529, A095263, A122432, A127894.

Flat(List([0..10],n->List([0..n],k->Binomial(n+2*k+2,n-k)))); # Muniru A Asiru, Apr 30 2018

[Binomial(n+2*k+2, n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Apr 29 2018

seq(seq(binomial(n+2*k+2,n-k),k=0..n),n=0..10); # Robert Israel, Apr 28 2015

Flatten@ Table[Binomial[n+2k-1, n-k], {n, 10}, {k, n}] (* Michael De Vlieger, Apr 27 2015 *)

for(n=0,10, for(k=0,n, print1(binomial(n+2*k+2, n-k), ", "))) \\ G. C. Greubel, Apr 29 2018

flatten([[binomial(n+2*k+2,n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 16 2021

T(n,k) = binomial(n+2*k+2, n-k).
Sum_{k=0..n} T(n, k) = A052529(n+1) (row sums).
Sum_{k=0..floor(n/2)} T(n-k, k) = A095263(n+1) (diagonal sums).
Recurrence: T(n+1, k+1) = Sum_{i = 0..n-k} binomial(i+2, 2)*T(n-i,k). - Peter Bala, Jul 22 2014
G.f.: 1/((1-x)^3-x*y). - Vladimir Kruchinin, Apr 27 2015

A055990 a(n) is its own 4th difference.

Original entry on oeis.org

1, 4, 14, 50, 181, 657, 2385, 8657, 31422, 114051, 413966, 1502555, 5453761, 19795288, 71850128, 260791401, 946583628, 3435774958, 12470688498, 45264335853, 164294064481, 596331286321, 2164478699633, 7856317702310, 28515747394555, 103502414271126

Offset: 1

Henry Bottomley, Jun 02 2000

Number of compositions of 4*n-2 into parts 1 and 4. - Seiichi Manyama, Feb 03 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-6,4,-1).

Cf. A055988, A055989, A055991 for the other differences of a(n). See A000079, A001906, A052529 for examples of sequences which are respectively their own first, second and third differences.

Cf. A003269.

I:=[1, 4, 14, 50]; [n le 4 select I[n] else 5*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 06 2012

CoefficientList[Series[(1-x)/(1-5*x+6*x^2-4*x^3+x^4),{x,0,30}],x] (* Vincenzo Librandi, Apr 06 2012 *)
LinearRecurrence[{5,-6,4,-1},{1,4,14,50},30] (* Harvey P. Dale, Oct 18 2015 *)

a(n):=sum((binomial(n+3*m+1,n-m-1)),m,0,n-1); /* Vladimir Kruchinin, Nov 18 2020 */

Vec((1-x)/(1-5*x+6*x^2-4*x^3+x^4)+O(x^99)) \\ Charles R Greathouse IV, Apr 06 2012

a(n) = 5*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) = a(n-1)+A055989(n) = A055991(n)-A055991(n-1) = A055988(n+1)-2*A055988(n)+A055988(n-1).
G.f.: x*(1-x)/(1-5*x+6*x^2-4*x^3+x^4). [Colin Barker, Apr 05 2012]
a(n) = Sum_{m=0..n-1} C(n+3m+1,n-m-1). - Vladimir Kruchinin, Nov 18 2020

More terms from James Sellers, Jun 05 2000

A365150 G.f. satisfies A(x) = 1 + xA(x)^2 / (1 - xA(x))^3.

Original entry on oeis.org

1, 1, 5, 26, 150, 925, 5967, 39772, 271758, 1893431, 13400897, 96078789, 696333585, 5093266409, 37549674939, 278739057687, 2081637677823, 15628794649931, 117897848681271, 893167062280029, 6792410218680749, 51835002735642287, 396821349652564273

Offset: 0

Seiichi Manyama, Aug 23 2023

nonn

Cf. A001003, A011270.
Cf. A365151, A365152.
Cf. A052529.

a(n, s=3, t=1) = sum(k=0, n, binomial(t*(n+k+1), k)*binomial(n+(s-1)*k-1, n-k)/(n+k+1));

If g.f. satisfies A(x) = ( 1 + x*A(x)^2 / (1 - x*A(x))^s )^t, then a(n) = Sum_{k=0..n} binomial(t*(n+k+1),k) * binomial(n+(s-1)*k-1,n-k)/(n+k+1).

G.f.: (1/x) * Series_Reversion( x*(1 - x/(1 - x)^3) ). - Seiichi Manyama, Sep 24 2024

A099239 Square array read by antidiagonals associated with sections of 1/(1-x-x^k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 8, 4, 1, 1, 16, 21, 13, 5, 1, 1, 32, 55, 41, 19, 6, 1, 1, 64, 144, 129, 69, 26, 7, 1, 1, 128, 377, 406, 250, 106, 34, 8, 1, 1, 256, 987, 1278, 907, 431, 153, 43, 9, 1, 1, 512, 2584, 4023, 3292, 1757, 686, 211, 53, 10, 1, 1, 1024, 6765, 12664, 11949, 7168, 3088, 1030, 281, 64, 11, 1

Offset: 0

Paul Barry, Oct 08 2004

Rows include A099242, A099253. Columns include A034856. Main diagonal is A099240. Sums of antidiagonals are A099241.

			Rows begin
  1, 1,  1,   1,   1, ...                               A000012;
  1, 2,  4,   8,  16, ...      1-section of 1/(1-x-x)   A000079;
  1, 3,  8,  21,  55, ....     bisection of 1/(1-x-x^2) A001906;
  1, 4, 13,  41, 129, ...     trisection of 1/(1-x-x^3) A052529; (essentially)
  1, 5, 19,  69, 250, ...  quadrisection of 1/(1-x-x^4) A055991;
  1, 6, 26, 106, 431, ...  quintisection of 1/(1-x-x^5) A079675; (essentially)

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

Cf. A034856, A099240, A099241, A099242, A099253.

Cf. A000079, A001906, A052529, A055991, A079675.

A099239:= func< n,k | (&+[Binomial(k*(n-k) -(k-1)*(j-1), j): j in [0..n-k]]) >;
[A099239(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 09 2021

T[n_, k_]:= Sum[Binomial[k*(n-k) - (k-1)*(j-1), j], {j,0,n-k}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 09 2021 *)

def A099239(n,k): return sum( binomial(k*(n-k) -(k-1)*(j-1), j) for j in (0..n-k) )
flatten([[A099239(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 09 2021

T(n, k) = Sum_{j=0..n} binomial(k*n -(k-1)*(j-1), j), n, k>=0. (square array)
T(n, k) = Sum_{j=0..n} binomial(k + (n-1)*(j+1), n*(j+1) -1), n>0. (square array)
T(n, k) = Sum_{j=0..n-k} binomial(k*(n-k) - (k-1)*(j-1), j). (number triangle)
Rows of the square array are generated by 1/((1-x)^k-x).
Rows satisfy a(n) = a(n-1) - Sum_{k=1..n} (-1)^(k^binomial(n, k)) * a(n-k).

A055989 a(n) is its own 4th difference.

Original entry on oeis.org

1, 3, 10, 36, 131, 476, 1728, 6272, 22765, 82629, 299915, 1088589, 3951206, 14341527, 52054840, 188941273, 685792227, 2489191330, 9034913540, 32793647355, 119029728628, 432037221840, 1568147413312, 5691839002677, 20659429692245, 74986666876571, 272175964826781

Offset: 1

Henry Bottomley, Jun 02 2000

Number of compositions of 4*n-3 into parts 1 and 4. - Seiichi Manyama, Feb 03 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-6,4,-1).

Cf. A055988, A055990, A055991 for the other differences of a(n). See A000079, A001906, A052529 for examples of sequences which are respectively their own first, second and third differences.

Cf. A003269.

I:=[1, 3, 10, 36]; [n le 4 select I[n] else 5*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 05 2012

CoefficientList[Series[(1-x)^2/(1-5*x+6*x^2-4*x^3+x^4),{x,0,40}],x] (* Vincenzo Librandi, Apr 05 2012 *)
LinearRecurrence[{5,-6,4,-1},{1,3,10,36},30] (* Harvey P. Dale, Jan 10 2014 *)

a(n) = 5*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) = a(n-1) + A055988(n) = A055990(n) - A055990(n-1) = A055991(n) - 2*A055991(n-1) + A055991(n-2).

G.f.: x*(1-x)^2/(1 - 5*x + 6*x^2 - 4*x^3 + x^4). - Colin Barker Apr 04 2012

More terms from James Sellers, Jun 05 2000

A192808 Constant term in the reduction of the polynomial (x^2 + 2)^n by x^3 -> x^2 + 2. See Comments.

Original entry on oeis.org

1, 2, 6, 26, 126, 618, 3022, 14746, 71902, 350538, 1708910, 8331130, 40615294, 198004778, 965298958, 4705957722, 22942154782, 111845982474, 545263681710, 2658231220538, 12959222223038, 63177890368490, 308000415667278, 1501542003033370

Offset: 0

Clark Kimberling, Jul 10 2011

For discussions of polynomial reduction, see A192232 and A192744.
If the reduction (x^2 + c)^n by x^3 -> x^2 + c is applied to the polynomials (x^2+c)^n for c=1 instead of c=2, the results are as follows:
A052554: constant terms,
A052529: coefficients of x,
A124820: coefficients of x^2.
Those three sequences satisfy the recurrence:
u(n) = 4*u(n-1) - 3*u(n-2) + u(n-3).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-12,8).

Cf. A192744, A192232, A192808 - A192811.

a:=[1,2,6];; for n in [4..25] do a[n]:=7*a[n-1]-12*a[n-2]+8*a[n-3]; od; Print(a); # Muniru A Asiru, Jan 02 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3) )); // G. C. Greubel, Jan 02 2019

q = x^3; s = x^2 + 2; z = 40;
p[n_, x_] := (x^2 + 2)^n;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192808 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192809 *)
u3 = Table[Coefficient[Part[t, n], x, 2], {n, 1, z}] (* A192810 *)
uu = u2/2  (* A192811 *)
LinearRecurrence[{7,-12,8}, {1,2,6}, 50] (* G. C. Greubel, Jan 02 2019 *)

my(x='x+O('x^30)); Vec((1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3)) \\ G. C. Greubel, Jan 02 2019

((1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 02 2019

a(n) = 7*a(n-1) - 12*a(n-2) + 8*a(n-3).

G.f.: (1-x)*(1-4*x)/(1-7*x+12*x^2-8*x^3). - Colin Barker, Jul 26 2012

A308946 Expansion of e.g.f. 1/(1 - x(1 + x/2)exp(x)).

Original entry on oeis.org

1, 1, 5, 30, 244, 2485, 30351, 432502, 7043660, 129050649, 2627117875, 58829021416, 1437117395946, 38032508860177, 1083932872119839, 33098858988564090, 1078083456543449416, 37309607437056658129, 1367138649165397662627, 52879280631976735387588

Offset: 0

Ilya Gutkovskiy, Jul 02 2019

nonn

Cf. A000217, A001793, A006153, A052529, A279361, A308861.

nmax = 19; CoefficientList[Series[1/(1 - x (1 + x/2) Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[k + 1, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

E.g.f.: 1 / (1 - Sum_{k>=1} (k*(k + 1)/2)*x^k/k!).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000217(k) * a(n-k).
a(n) ~ n! * (2 + r) / ((2 + 4*r + r^2) * r^n), where r = 0.49122518354447387971550543450091640839121607... is the root of the equation  exp(r)*r*(2 + r) = 2. - Vaclav Kotesovec, Aug 09 2021

A382615 Expansion of 1/(1 - x/(1 - x)^3)^3.

Original entry on oeis.org

1, 3, 15, 64, 261, 1032, 3982, 15066, 56094, 206068, 748452, 2691966, 9600233, 33982197, 119495229, 417724302, 1452550371, 5026878774, 17321417650, 59450099958, 203306331429, 692955932103, 2354664287943, 7978488379398, 26963061909228, 90897971951727

Offset: 0

Seiichi Manyama, Mar 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (12,-57,139,-195,174,-102,39,-9,1).

Cf. A000217, A058396, A290918.
Cf. A052529, A382616.
Cf. A290917.

R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x/(1 - x)^3)^3; seq := [ Coefficient(f, n) : n in [0..30] ]; seq;// Vincenzo Librandi, Apr 02 2025

Table[Sum[Binomial[k+2,2]*Binomial[n+2*k-1,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Apr 02 2025 *)

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

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

a(n) = 12*a(n-1) - 57*a(n-2) + 139*a(n-3) - 195*a(n-4) + 174*a(n-5) - 102*a(n-6) + 39*a(n-7) - 9*a(n-8) + a(n-9) for n > 9.

A382616 Expansion of 1/(1 - x/(1 - x)^3)^2.

Original entry on oeis.org

1, 2, 9, 34, 124, 444, 1567, 5466, 18885, 64732, 220403, 746166, 2513678, 8431650, 28175256, 93834240, 311565255, 1031723268, 3408137644, 11233323692, 36950587185, 121319416734, 397649266199, 1301332828086, 4252515425757, 13877722224278, 45232020345642

Offset: 0

Seiichi Manyama, Mar 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (8,-22,26,-17,6,-1).

Cf. A045623, A290917.

Cf. A052529, A382615.

R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x/(1 - x)^3)^2; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 02 2025

Table[Sum[(k+1)*Binomial[n+2*k-1,n-k],{k,0,n}],{n,0,26}] (* Vincenzo Librandi, Apr 02 2025 *)

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

a(n) = Sum_{k=0..n} (k+1) * binomial(n+2*k-1,n-k).
a(n) = 8*a(n-1) - 22*a(n-2) + 26*a(n-3) - 17*a(n-4) + 6*a(n-5) - a(n-6) for n > 6.
G.f.: (x-1)^6/(x^3-3*x^2+4*x-1)^2.

A378566 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(n+2*k-1,n-k).

Original entry on oeis.org

1, 1, 9, 64, 465, 3456, 26082, 199060, 1532313, 11875015, 92528414, 724187982, 5689127886, 44834549501, 354289977750, 2806262293824, 22273793685609, 177113634045858, 1410633764438967, 11251419724586850, 89860413370562730, 718528004169570925

Offset: 0

Seiichi Manyama, Dec 01 2024

nonn

Cf. A002002, A378565, A378567.

Cf. A052529, A362088, A365150.

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

a(n) = [x^n] 1/(1 - x/(1 - x)^3)^n.

cp's OEIS Frontend

A127893 Riordan array (1/(1-x)^3, x/(1-x)^3).

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A055990 a(n) is its own 4th difference.

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A365150 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 - x*A(x))^3.

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A099239 Square array read by antidiagonals associated with sections of 1/(1-x-x^k).

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A055989 a(n) is its own 4th difference.

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Magma

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A192808 Constant term in the reduction of the polynomial (x^2 + 2)^n by x^3 -> x^2 + 2. See Comments.

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A308946 Expansion of e.g.f. 1/(1 - x*(1 + x/2)*exp(x)).

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A365150 G.f. satisfies A(x) = 1 + xA(x)^2 / (1 - xA(x))^3.

A308946 Expansion of e.g.f. 1/(1 - x(1 + x/2)exp(x)).