A001972 -id:A001972 - cp's OEIS frontend

A349839 Triangle T(n,k) built by placing all ones on the left edge, [1,0,0,0] repeated on the right edge, and filling the body using the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 1, 1, 4, 6, 4, 2, 0, 1, 5, 10, 10, 6, 2, 0, 1, 6, 15, 20, 16, 8, 2, 0, 1, 7, 21, 35, 36, 24, 10, 2, 1, 1, 8, 28, 56, 71, 60, 34, 12, 3, 0, 1, 9, 36, 84, 127, 131, 94, 46, 15, 3, 0, 1, 10, 45, 120, 211, 258, 225, 140, 61, 18, 3, 0, 1, 11, 55, 165, 331, 469, 483, 365, 201, 79, 21, 3, 1

Offset: 0

Michael A. Allen, Dec 01 2021

This is the m=4 member in the sequence of triangles A007318, A059259, A118923, A349839, A349841 which have all ones on the left side, ones separated by m-1 zeros on the other side, and whose interiors obey Pascal's recurrence.
T(n,k) is the (n,n-k)-th entry of the (1/(1-x^4),x/(1-x)) Riordan array.
For n>0, T(n,n-1) = A008621(n-1).
For n>1, T(n,n-2) = A001972(n-2).
For n>2, T(n,n-3) = A122046(n).
Sums of rows give A115451.
Sums of antidiagonals give A349840.

			Triangle begins:
  1;
  1,   0;
  1,   1,   0;
  1,   2,   1,   0;
  1,   3,   3,   1,   1;
  1,   4,   6,   4,   2,   0;
  1,   5,  10,  10,   6,   2,   0;
  1,   6,  15,  20,  16,   8,   2,   0;
  1,   7,  21,  35,  36,  24,  10,   2,   1;
  1,   8,  28,  56,  71,  60,  34,  12,   3,   0;
  1,   9,  36,  84, 127, 131,  94,  46,  15,   3,   0;
  1,  10,  45, 120, 211, 258, 225, 140,  61,  18,   3,   0;
  1,  11,  55, 165, 331, 469, 483, 365, 201,  79,  21,   3,   1;

Michael A. Allen and Kenneth Edwards, On Two Families of Generalizations of Pascal's Triangle, J. Int. Seq. 25 (2022) Article 22.7.1.

Other members of sequence of triangles: A007318, A059259, A118923, A349841.
Columns: A000012, A001477, A000217, A000292, A145126, A051745.
Diagonals: A121262, A008621, A001972, A122046.

Flatten[Table[CoefficientList[Series[(1-x*y)/((1-(x*y)^4)(1 - x - x*y)), {x, 0, 24}, {y, 0, 12}], {x, y}][[n+1,k+1]],{n,0,12},{k,0,n}]]

G.f.: (1-x*y)/((1-(x*y)^4)(1-x-x*y)) in the sense that T(n,k) is the coefficient of x^n*y^k in the series expansion of the g.f.
T(n,0) = 1.
T(n,n) = delta(n mod 4,0).
T(n,1) = n-1 for n>0.
T(n,2) = (n-1)*(n-2)/2 for n>1.
T(n,3) = (n-1)*(n-2)*(n-3)/6 for n>2.
T(n,4) = C(n-1,4) + 1 for n>3.
T(n,5) = C(n-1,5) + n - 5 for n>4.
For 0 <= k < n, T(n,k) = (n-k)*Sum_{j=0..floor(k/4)} binomial(n-4*j,n-k)/(n-4*j).
The g.f. of the n-th subdiagonal is 1/((1-x^4)(1-x)^n).

A008730 Molien series 1/((1-x)^2(1-x^12)) for 3-dimensional group [2,n] = 22n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 186, 192, 198, 204

Offset: 0

N. J. A. Sloane

			..1....2....3....4....5....6....7....8....9...10...11...12
.14...16...18...20...22...24...26...28...30...32...34...36
.39...42...45...48...51...54...57...60...63...66...69...72
.76...80...84...88...92...96..100..104..108..112..116..120
125..130..135..140..145..150..155..160..165..170..175..180
186..192..198..204..210..216..222..228..234..240..246..252
259..266..273..280..287..294..301..308..315..322..329..336
344..352..360..368..376..384..392..400..408..416..424..432
441..450..459..468..477..486..495..504..513..522..531..540
550..560..570..580..590..600..610..620..630..640..650..660
...
The columns are: A051866, A139267, A094159, A033579, A049452, A033581, A049453, A033580, A195319, A202804, A211014, A049598
- _Philippe Deléham_, Apr 03 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 195
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1).

Cf. A001840, A001972, A008724, A008725, A008726, A008727, A008728, A008729, A008732. - Vladimir Joseph Stephan Orlovsky, Mar 14 2010

Cf. A221912

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/((1-x)^2*(1-x^12)) )); // G. C. Greubel, Jul 30 2019

seq(coeff(series(1/(1-x)^2/(1-x^12), x, n+1), x, n), n=0..80);

CoefficientList[Series[1/((1-x)^2*(1-x^12)), {x,0,70}], x] (* Vincenzo Librandi, Jun 11 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{1,2,3,4,5,6,7,8,9,10,11,12,14,16},70] (* Harvey P. Dale, Jan 01 2024 *)

my(x='x+O('x^70)); Vec(1/((1-x)^2*(1-x^12))) \\ G. C. Greubel, Jul 30 2019

(1/((1-x)^2*(1-x^12))).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 30 2019

G.f. 1/( (1-x)^3 * (1+x) *(1+x+x^2) *(1-x+x^2) * (1+x^2) *(1-x^2+x^4)). - R. J. Mathar, Aug 11 2021
From Mitch Harris, Sep 08 2008: (Start)
a(n) = Sum_{j=0..n+12} floor(j/12).
a(n-12) = (1/2)*floor(n/12)*(2*n - 10 - 12*floor(n/12)). (End)
a(n) = A221912(n+12). - Philippe Deléham, Apr 03 2013

More terms from Vladimir Joseph Stephan Orlovsky, Mar 14 2010

A189074 Irregular triangle read by rows: T(n,k) = number of compositions of n with k inversions (n >= 0, 0 <= k <= floor(n^2/8)).

Original entry on oeis.org

1, 1, 2, 3, 1, 5, 2, 1, 7, 5, 3, 1, 11, 8, 7, 4, 2, 15, 15, 14, 10, 6, 3, 1, 22, 23, 26, 21, 17, 10, 6, 2, 1, 30, 37, 44, 42, 36, 27, 19, 11, 6, 3, 1, 42, 55, 73, 74, 73, 60, 50, 34, 24, 13, 8, 4, 2, 56, 83, 115, 128, 133, 123, 109, 87, 68, 48, 32, 20, 12, 6, 3, 1, 77, 118, 177, 209, 235, 230, 223, 192, 166, 129, 100, 70, 51, 31, 20, 11, 6, 2, 1

Offset: 0

N. J. A. Sloane, Apr 16 2011

Row sums are powers of 2.

The Heubach et al. reference has a table for n <= 12.

			T(4,0) = 5: [4], [1,3], [2,2], [1,1,2], [1,1,1,1] - all partitions of 4.
T(5,2) = 3: [2,2,1], [3,1,1], [1,2,1,1].
T(6,4) = 2: [2,2,1,1], [2,1,1,1,1].
Triangle begins:
1
1
2
3   1
5   2  1
7   5  3  1
11  8  7  4  2
15 15 14 10  6  3 1
22 23 26 21 17 10 6 2 1
...

Alois P. Heinz, Rows n = 0..25, flattened
S. Heubach, A. Knopfmacher, M. E. Mays and A. Munagi, Inversions in Compositions of Integers, to appear in Quaestiones Mathematicae.

The first two columns are A000041 and A058884 (for n>0). Lengths of rows are given by 1+A001972(n-3). Row sums are A011782.

T:= proc(n) option remember; local b, p;
      b:=proc(m, i, l)
           if m=0 then p(i):= p(i)+1
         else seq(b(m-h, i+nops(select(j->jAlois P. Heinz, Apr 17 2011

T[n_] := T[n] = Module[{b, p}, b[m_, i_, l_List] := If[m == 0, p[i] = p[i] + 1, Table[b[m-h, i+Length[Select[ l, #]=0; b[n, 0, {}]; Table[p[i], {i, 0, Floor[n^2/8]}]]; Table[ T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover, Jan 17 2016, after Alois P. Heinz *)

A194200 [sum{(k*e) : 1<=k<=n}], where [ ]=floor, ( )=fractional part.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 7, 8, 9, 9, 9, 10, 11, 11, 12, 13, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 19, 20, 20, 21, 22, 22, 22, 23, 23, 23, 24, 25, 25, 25, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 34, 34, 35, 36, 37

Offset: 1

Clark Kimberling, Aug 19 2011

nonn

The defining [sum] is equivalent to
...
a(n)=[n(n+1)r/2]-sum{[k*r] : 1<=k<=n},
...
where []=floor and r=sqrt(2).  Let s(n) denote the n-th partial sum of the sequence a; then the difference sequence d defined by d(n)=s(n+1)-s(n) gives the runlengths of a.
...
Examples:
...
r...........a........s....
1/2......A002265...A001972
1/3......A002264...A001840
2/3......A002264...A001840
1/4......A194220...A194221
1/5......A194222...A118015
2/5......A057354...A011858
3/5......A194222...A118015
4/5......A057354...A011858
1/6......A194223...A194224
3/7......A057357...A194229
1/8......A194235...A194236
3/8......A194237...A194238
sqrt(2)..A194161...A194162
sqrt(3)..A194163...A194164
sqrt(5)..A194169...A194170
sqrt(6)..A194173...A194174
tau......A194165...A194166; tau=(1+sqrt(5))/2
e........A194200...A194201
2e.......A194202...A194203
e/2......A194204...A194205
pi.......A194206...A194207

			a(5)=[(e)+(2e)+(3e)+4(e)+5(e)]
    =[.718+.436+.154+.873+.591]
    =[2.77423]=2.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A194201.

r = E;
a[n_] := Floor[Sum[FractionalPart[k*r], {k, 1, n}]]
Table[a[n], {n, 1, 90}]  (* A194200 *)
s[n_] := Sum[a[k], {k, 1, n}]
Table[s[n], {n, 1, 100}] (* A194201 *)

A078529 Exponent sequence for a bilinear recursive sequence.

Original entry on oeis.org

3, 1, 0, 0, 0, 0, 0, 1, 2, 3, 4, 6, 9, 10, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 51, 55, 60, 66, 72, 78, 84, 91, 98, 105, 112, 120, 129, 136, 144, 153, 162, 171, 180, 190, 200, 210, 220, 231, 243, 253, 264, 276, 288, 300, 312, 325, 338, 351, 364, 378, 393, 406, 420

Offset: 0

Michael Somos, Nov 25 2002

			3 + x + x^7 + 2*x^8 + 3*x^9 + 4*x^10 + 6*x^11 + 9*x^12 + 10*x^13 + ...

Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A001972, A078530.

LinearRecurrence[{2,-1,0,0,0,0,0,0,0,0,0,1,-2,1},{3,1,0,0,0,0,0,1,2,3,4,6,9,10},70] (* Harvey P. Dale, May 27 2017 *)

PARI
```
{a(n) = (n%12==0) + (n-4)^2\8}
```

G.f.: (3 - 5*x + x^2 + x^3 + x^7 + x^11 - 2*x^12 + 3*x^13) / ((1 - x)^2 * (1 - x^12)).

a(8-n) - a(n) = -1 if n == 0 (mod 12), +1 if n == 8 (mod 12), 0 otherwise.

A182568 a(n) = 2floor(n/4)(n - 2*(1 + floor(n/4))).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 4, 6, 8, 12, 16, 20, 24, 30, 36, 42, 48, 56, 64, 72, 80, 90, 100, 110, 120, 132, 144, 156, 168, 182, 196, 210, 224, 240, 256, 272, 288, 306, 324, 342, 360, 380, 400, 420, 440, 462, 484, 506, 528, 552, 576, 600, 624, 650, 676, 702, 728, 756, 784, 812, 840, 870, 900, 930, 960, 992, 1024, 1056, 1088, 1122, 1156, 1190, 1224, 1260, 1296, 1332, 1368

Offset: 0

N. J. A. Sloane, May 05 2012

Pak Tung Ho, The toroidal crossing number of K_{4,n}, Discrete Math. 309 (2009), no. 10, 3238--3248. MR2526742(2010i:05088).
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
Eric Weisstein's World of Mathematics, Toroidal Crossing Number
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Cf. A001972, A130519.

Table[2 Floor[n/4] (n - 2 (1 + Floor[n/4])), {n, 0, 20}] (* or *)
Table[(5 - (-1)^n + 2 (n - 4) n - 4 Cos[n Pi/2])/8, {n, 0, 20}] (* or *)
Table[(5 - (-1)^n - 2 (-I)^n - 2 I^n - 8 n + 2 n^2)/8, {n, 0, 20}] (* or *)
LinearRecurrence[{2, -1, 0, 1, -2, 1}, {0, 0, 0, 0, 0, 2}, 80] (* or *)
CoefficientList[Series[-2 x^5/((-1 + x)^3 (1 + x + x^2 + x^3)), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 11 2018 *)

From R. J. Mathar, Jun 28 2012: (Start)
G.f. -2*x^5 / ( (x + 1)*(x^2 + 1)*(x - 1)^3 ).
a(n) = 2*A001972(n-5) = 2*A130519(n-1). (End)
a(n) = 2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6). - Eric W. Weisstein, Sep 11 2018

A025842 Expansion of 1/((1-x^3)(1-x^6)(1-x^8)).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 2, 0, 1, 2, 0, 1, 3, 0, 2, 3, 1, 2, 4, 1, 3, 4, 2, 3, 6, 2, 4, 6, 3, 4, 8, 3, 6, 8, 4, 6, 10, 4, 8, 10, 6, 8, 12, 6, 10, 12, 8, 10, 15, 8, 12, 15, 10, 12, 18, 10, 15, 18, 12, 15, 21, 12, 18, 21, 15, 18, 24, 15, 21

Offset: 0

N. J. A. Sloane

a(n) is the number of partitions of n into parts 3, 6, and 8. - Michel Marcus, Jun 30 2025

Index entries for linear recurrences with constant coefficients, signature (0,0,1,0,0,1,0,1,-1,0,-1,0,0,-1,0,0,1).

Cf. A001972, A004526.

CoefficientList[Series[1/((1-x^3)(1-x^6)(1-x^8)),{x,0,70}],x] (* Harvey P. Dale, Jan 25 2012 *)

a(3*n) = a(3*n+8) = a(3*n+16) = A001972(A004526(n)). - Hoang Xuan Thanh, Jun 25 2025

cp's OEIS Frontend

A349839 Triangle T(n,k) built by placing all ones on the left edge, [1,0,0,0] repeated on the right edge, and filling the body using the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

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A008730 Molien series 1/((1-x)^2*(1-x^12)) for 3-dimensional group [2,n] = *22n.

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A189074 Irregular triangle read by rows: T(n,k) = number of compositions of n with k inversions (n >= 0, 0 <= k <= floor(n^2/8)).

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A194200 [sum{(k*e) : 1<=k<=n}], where [ ]=floor, ( )=fractional part.

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A078529 Exponent sequence for a bilinear recursive sequence.

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A182568 a(n) = 2*floor(n/4)*(n - 2*(1 + floor(n/4))).

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A025842 Expansion of 1/((1-x^3)*(1-x^6)*(1-x^8)).

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A008730 Molien series 1/((1-x)^2(1-x^12)) for 3-dimensional group [2,n] = 22n.

A182568 a(n) = 2floor(n/4)(n - 2*(1 + floor(n/4))).

A025842 Expansion of 1/((1-x^3)(1-x^6)(1-x^8)).