Original entry on oeis.org
1, 1, 2, 1, 4, 3, 8, 7, 1, 16, 15, 5, 32, 31, 16, 1, 64, 63, 42, 7, 128, 127, 99, 29, 1256, 255, 219, 93, 9, 512, 511, 466, 256, 46, 1, 1024, 1023, 968, 638, 176, 11, 2048, 2047, 1981, 1486, 562, 67, 1
Offset: 0
First few rows of the triangle =
1;
1;
2, 1;
4, 3;
8, 7, 1;
16, 15, 5;
32, 31, 16, 1;
64, 63, 42, 7;
128, 127, 99, 29, 1;
256, 255, 219, 93, 9;
512, 511, 466, 256, 46, 1;
1024, 1023, 968, 638, 176, 11;
2048, 2047, 1981, 1486, 562, 67, 1;
...
Original entry on oeis.org
1, 1, 1, 1, 1, 3, 1, 6, 2, 1, 10, 10, 1, 15, 30, 4, 1, 21, 70, 28, 1, 28, 140, 112, 9, 1, 36, 252, 336, 81, 1, 45, 420, 840, 405, 21, 1, 55, 660, 1848, 1485, 231, 1, 66, 990, 3696, 4455, 1386, 50
Offset: 0
First few rows of the triangle =
1;
1;
1, 1;
1, 3;
1, 6, 2;
1, 10, 10;
1, 15, 30, 4;
1, 21, 70, 28;
1, 28, 140, 112, 9;
1, 36, 252, 336, 81;
1, 45, 420, 840, 405, 21;
1, 55, 660, 1848, 1485, 231;
1, 66, 990, 3696, 4455, 1386, 50;
...
A208510
Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section.
Original entry on oeis.org
1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 7, 9, 5, 1, 1, 9, 16, 14, 6, 1, 1, 11, 25, 30, 20, 7, 1, 1, 13, 36, 55, 50, 27, 8, 1, 1, 15, 49, 91, 105, 77, 35, 9, 1, 1, 17, 64, 140, 196, 182, 112, 44, 10, 1, 1, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 1, 21, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1
Offset: 1
First five rows:
1
1...1
1...3...1
1...5...4...1
1...7...9...5...1
First five polynomials u(n,x):
1
1 + x
1 + 3x + x^2
1 + 5x + 4x^2 + x^3
1 + 7x + 9x^2 + 5x^3 + x^4
-
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 *)
-
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(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017
A000579
Figurate numbers or binomial coefficients C(n,6).
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 1, 7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 376740, 475020, 593775, 736281, 906192, 1107568, 1344904, 1623160, 1947792, 2324784, 2760681, 3262623
Offset: 0
a(9) = 84 = (1, 3, 3, 1) dot (1, 6, 15, 20) = (1 + 18 + 45 + 20). - _Gary W. Adamson_, Aug 02 2008
G.f. = x^6 + 7*x^7 + 28*x^8 + 84*x^9 + 210*x^10 + 462*x^11 + 924*x^12 + ...
For A = {1,2,3,4,5,6} subsets with 5 elements are {1,2,3,4,5}, {1,2,3,4,6}, {1,2,3,5,6}, {1,2,4,5,6}, {1,3,4,5,6}, {2,3,4,5,6}. Sum of 2 smallest elements of each subset: a(6) = (1+2) + (1+2) + (1+2) + (1+2) + (1+3) + (2+3) = 21 = 3*C(6+1,6) = 3*A000579(6+1). - _Serhat Bulut_, Oktay Erkan Temizkan, Mar 13 2015
a(7) = 7 from the seven independent components of an antisymmetric tensor A of rank 6 and dimension 7: A(1,2,3,4,5,6), A(1,2,3,4,5,7), A(1,2,3,4,6,7), A(1,2,3,5,6,7) A(1,2,4,5,6,7), A(1,2,3,5,6,7) and A(2,3,4,5,6,7). See a Dec 10 2015 comment. - _Wolfdieter Lang_, Dec 10 2015
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
- A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 196.
- L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 7.
- J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Charles W. Trigg: Mathematical Quickies. New York: Dover Publications, Inc., 1985, p. 11, #32
- T. D. Noe, Table of n, a(n) for n = 0..1000
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
- Matthias Beck and Serkan Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
- Serhat Bulut and Oktay Erkan Temizkan, Subset Sum Problem
- Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
- Philippe A. J. G. Chevalier, On the discrete geometry of physical quantities, Preprint, 2012.
- Philippe A. J. G. Chevalier, On a Mathematical Method for Discovering Relations Between Physical Quantities: a Photonics Case Study, Slides from a talk presented at ICOL2014.
- Philippe A. J. G. Chevalier, A "table of Mendeleev" for physical quantities?, Slides from a talk, May 14 2014, Leuven, Belgium.
- Philippe A. J. G. Chevalier, Dimensional exploration techniques for photonics, Slides of a talk, 2016.
- Ömür Deveci and Anthony G. Shannon, Some aspects of Neyman triangles and Delannoy arrays, Mathematica Montisnigri (2021) Vol. L, 36-43.
- Michael B. Green, Stephen D. Miller, and Pierre Vanhove, Small representations, string instantons, and Fourier modes of Eisenstein series, arXiv:1111.2983 [hep-th], 2011-2013.
- Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 4.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 256
- Milan Janjic, Two Enumerative Functions
- Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
- Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 13, 15.
- Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
- Leo Moser, Quicky 87, Mathematics Magazine, 26 (March 1953), p. 226.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000.
- Hermann Stamm-Wilbrandt, Sum of Pascal's triangle reciprocals [Cached copy from the Wayback Machine]
- Eric Weisstein's World of Mathematics, Composition
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
Cf.
A053135,
A053128,
A000580 (partial sums),
A000581,
A000582,
A000217,
A000292,
A000332,
A000389 (first differences),
A104712 (fifth column, k=6).
-
[Binomial(n,6) : n in [0..50]]; // Wesley Ivan Hurt, Jul 13 2014
-
A000579 := n->binomial(n,6);
ZL := [S, {S=Prod(B,B,B,B,B,B,B), B=Set(Z, 1 <= card)}, unlabeled]: seq(combstruct[count](ZL, size=n), n=7..40); # Zerinvary Lajos, Mar 13 2007
A000579:=-1/(z-1)**7; # Simon Plouffe in his 1992 dissertation, referring to offset 0.
seq(binomial(n,6),n=0..33); # Zerinvary Lajos, Jun 16 2008
G(x):=x^6*exp(x): f[0]:=G(x): for n from 1 to 39 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n]/6!,n=6..39); # Zerinvary Lajos, Apr 05 2009
-
Table[Binomial[n, 6], {n, 6, 50}] (* Stefan Steinerberger, Apr 02 2006 *)
Table[n(n - 1)(n - 2)(n - 3)(n - 4)(n - 5)/720, {n, 0, 100}] (* Artur Jasinski, Dec 02 2007 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,0,0,0,1},50] (* Harvey P. Dale, Dec 30 2012 *)
CoefficientList[ Series[ -7x^6/(x-1)^7,{x, 0, 35}], x]/7 (* Robert G. Wilson v, Jan 29 2015 *)
-
a(n)=binomial(n,6) \\ Charles R Greathouse IV, Nov 20 2012
-
A000579_list, m = [], [1, -5, 10, -10, 5, -1, 0]
for _ in range(10**2):
A000579_list.append(m[-1])
for i in range(6):
m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016
Some formulas that referred to other offsets corrected by
R. J. Mathar, Jul 07 2009
I changed the offset to 0. This will require some further adjustments to the formulas. -
N. J. A. Sloane, Aug 01 2010
Shevelev comment inserted and further adaptations to offset by
R. J. Mathar, Aug 03 2010
A098158
Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 1, 6, 1, 0, 0, 0, 5, 10, 1, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0, 0, 1, 66, 495, 924
Offset: 0
Rows begin
1;
0, 1;
0, 1, 1;
0, 0, 3, 1;
0, 0, 1, 6, 1;
-
Flat(List([0..12], n-> List([0..n], k-> Binomial(n, 2*(n-k)) ))); # G. C. Greubel, Aug 01 2019
-
[Binomial(n, 2*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019
-
Table[Binomial[n, 2*(n-k)], {n,0,12}, {k,0,n}]//Flatten (* Michael De Vlieger, Oct 12 2016 *)
-
{T(n,k)=polcoeff(polcoeff((1-x*y)/((1-x*y)^2-x^2*y)+x*O(x^n), n, x) + y*O(y^k),k,y)} (Hanna)
-
T(n,k) = binomial(n, 2*(n-k));
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Aug 01 2019
-
[[binomial(n, 2*(n-k)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019
A034851
Rows of Losanitsch's triangle T(n, k), n >= 0, 0 <= k <= n.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 9, 10, 9, 3, 1, 1, 4, 12, 19, 19, 12, 4, 1, 1, 4, 16, 28, 38, 28, 16, 4, 1, 1, 5, 20, 44, 66, 66, 44, 20, 5, 1, 1, 5, 25, 60, 110, 126, 110, 60, 25, 5, 1, 1, 6, 30, 85, 170, 236, 236, 170, 85, 30, 6, 1, 1, 6, 36, 110, 255
Offset: 0
Triangle begins
1;
1, 1;
1, 1, 1;
1, 2, 2, 1;
1, 2, 4, 2, 1;
1, 3, 6, 6, 3, 1;
1, 3, 9, 10, 9, 3, 1;
1, 4, 12, 19, 19, 12, 4, 1;
1, 4, 16, 28, 38, 28, 16, 4, 1;
1, 5, 20, 44, 66, 66, 44, 20, 5, 1;
- Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
- F. Al-Kharousi, R. Kehinde, and A. Umar, Combinatorial results for certain semigroups of partial isometries of a finite chain, The Australasian Journal of Combinatorics, Vol. 58, No. 3 (2014), pp. 363-375.
- Tewodros Amdeberhan, Mahir Bilen Can and Victor H. Moll, Broken bracelets, Molien series, paraffin wax and the elliptic curve of conductor 48, SIAM Journal of Discrete Math., Vol. 25, No. 4 (2011), p. 1843-1859; arXiv preprint, arXiv:1106.4693 [math.CO], 2011. See Theorem 2.8.
- Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.
- Sahir Gill, Bounds for Region Containing All Zeros of a Complex Polynomial, International Journal of Mathematical Analysis Vol. 12, No. 7 (2018), pp. 325-333.
- Stephen G. Hartke and A. J. Radcliffe, Signatures of Strings, Annals of Combinatorics, Vol. 17, No. 1 (March, 2013), pp. 131-150.
- Rethinasamy K. Kittappa, Combinatorial enumeration of rectangular kolam designs of the Tamil land, Abstracts Amer. Math. Soc., Vol. 29, No. 1 (2008), p. 24 (Abstract 1035-05-543).
- Wolfdieter Lang, Illustration of initial rows of triangle.
- S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. Vol. 30 (1897), pp. 1917-1926.
- S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber., Vol. 30 (1897), pp. 1917-1926. (Annotated scanned copy)
- Ministry of Foreign Affairs of Serbia, List of the Ministers for Foreign Affairs Since the Forming of the First Government in 1811-Sima Lozanic.
- Jesse Pajwani, Herman Rohrbach, and Anna M. Viergever, Compactly supported A^1-Euler characteristics of symmetric powers of cellular varieties, arXiv:2404.08486 [math.AG], 2024. See p. 15.
- N. J. A. Sloane, Classic Sequences.
- Eric Weisstein's World of Mathematics, Losanitsch's Triangle.
- Wikipedia, Sima Lozanic.
- Index entries for sequences related to trees
Triangle sums (see the comments):
A005418 (Row),
A011782 (Related to Row2),
A102526 (Related to Kn11, Kn12, Kn13, Kn21, Kn22, Kn23),
A005207 (Kn3, Kn4),
A005418 (Fi1, Fi2),
A102543 (Ca1, Ca2),
A192928 (Gi1, Gi2),
A005683 (Ze3, Ze4).
Sums of squares of terms in rows equal
A211208.
-
a034851 n k = a034851_row n !! k
a034851_row 0 = [1]
a034851_row 1 = [1,1]
a034851_row n = zipWith (-) (zipWith (+) ([0] ++ losa) (losa ++ [0]))
([0] ++ a204293_row (n-2) ++ [0])
where losa = a034851_row (n-1)
a034851_tabl = map a034851_row [0..]
-- Reinhard Zumkeller, Jan 14 2012
-
A034851 := proc(n,k) option remember; local t; if k = 0 or k = n then return(1) fi; if n mod 2 = 0 and k mod 2 = 1 then t := binomial(n/2-1,(k-1)/2) else t := 0; fi; A034851(n-1,k-1)+A034851(n-1,k)-t; end: seq(seq(A034851(n, k), k=0..n), n=0..11);
-
t[n_?EvenQ, k_?OddQ] := Binomial[n, k]/2; t[n_, k_] := (Binomial[n, k] + Binomial[Quotient[n, 2], Quotient[k, 2]])/2; Flatten[Table[t[n, k], {n, 0, 12}, {k, 0, n}]](* Jean-François Alcover, Feb 07 2012, after PARI *)
-
{T(n, k) = (1/2) *(binomial(n, k) + binomial(n%2, k%2) * binomial(n\2, k\2))}; /* Michael Somos, Oct 20 1999 */
A086645
Triangle read by rows: T(n, k) = binomial(2n, 2k).
Original entry on oeis.org
1, 1, 1, 1, 6, 1, 1, 15, 15, 1, 1, 28, 70, 28, 1, 1, 45, 210, 210, 45, 1, 1, 66, 495, 924, 495, 66, 1, 1, 91, 1001, 3003, 3003, 1001, 91, 1, 1, 120, 1820, 8008, 12870, 8008, 1820, 120, 1, 1, 153, 3060, 18564, 43758, 43758, 18564, 3060, 153, 1, 1, 190, 4845, 38760
Offset: 0
From _Peter Bala_, Oct 23 2008: (Start)
The triangle begins
n\k|..0.....1.....2.....3.....4.....5.....6
===========================================
0..|..1
1..|..1.....1
2..|..1.....6.....1
3..|..1....15....15.....1
4..|..1....28....70....28.....1
5..|..1....45...210...210....45.....1
6..|..1....66...495...924...495....66.....1
...
(End)
From _Peter Bala_, Aug 06 2013: (Start)
Viewed as the generalized Riordan array (cosh(sqrt(y)), y) with respect to the sequence (2*n)! the column generating functions begin
1st col: cosh(sqrt(y)) = 1 + y/2! + y^2/4! + y^3/6! + y^4/8! + ....
2nd col: 1/2!*y*cosh(sqrt(y)) = y/2! + 6*y^2/4! + 15*y^3/6! + 28*y^4/8! + ....
3rd col: 1/4!*y^2*cosh(sqrt(y)) = y^2/4! + 15*y^3/6! + 70*y^4/8! + 210*y^5/10! + .... (End)
- A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 224.
- Indranil Ghosh, Rows 0.. 120 of triangle, flattened
- F. Ardila, M. Beck, S. Hosten, J. Pfeifle and K. Seashore, Root polytopes and growth series of root lattices arXiv:0809.5123 [math.CO], 2008.
- H. Chan, S. Cooper and P. Toh, The 26th power of Dedekind's eta function Advances in Mathematics, 207 (2006) 532-543.
- T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras, 2012.
- T. Copeland, Skipping over Dimensions, Juggling Zeros in the Matrix, 2020.
- E. Lucas, Théorie des fonctions numériques simplement périodiques, Baltimore, 1878. See page 145 equation (153).
- W. Wang and T. Wang, Generalized Riordan array, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
-
/* As triangle: */ [[Binomial(2*n, 2*k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Dec 14 2016
-
A086645:=(n,k)->binomial(2*n,2*k): seq(seq(A086645(n,k),k=0..n),n=0..12);
-
Table[Binomial[2 n, 2 k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 13 2016 *)
-
create_list(binomial(2*n,2*k),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */
-
{T(n, k) = binomial(2*n, 2*k)};
-
{T(n, k) = sum( i=0, min(k, n-k), 4^i * binomial(n, 2*i) * binomial(n - 2*i, k-i))}; /* Michael Somos, May 26 2005 */
A384175
Number of subsets of {1..n} with all distinct lengths of maximal runs (increasing by 1).
Original entry on oeis.org
1, 2, 4, 7, 13, 24, 44, 77, 135, 236, 412, 713, 1215, 2048, 3434, 5739, 9559, 15850, 26086, 42605, 69133, 111634, 179602, 288069, 460553, 733370, 1162356, 1833371, 2878621, 4501856, 7016844, 10905449, 16904399, 26132460, 40279108, 61885621, 94766071, 144637928
Offset: 0
The subset {2,3,5,6,7,9} has maximal runs ((2,3),(5,6,7),(9)), with lengths (2,3,1), so is counted under a(9).
The a(0) = 1 through a(4) = 13 subsets:
{} {} {} {} {}
{1} {1} {1} {1}
{2} {2} {2}
{1,2} {3} {3}
{1,2} {4}
{2,3} {1,2}
{1,2,3} {2,3}
{3,4}
{1,2,3}
{1,2,4}
{1,3,4}
{2,3,4}
{1,2,3,4}
For equal instead of distinct lengths we have
A243815.
These subsets are ranked by
A328592.
The complement is counted by
A384176.
For permutations instead of subsets we have
A384891, equal instead of distinct
A384892.
A034839 counts subsets by number of maximal runs, for strict partitions
A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement
A336866.
Cf.
A000009,
A010027,
A044813,
A047993,
A242882,
A325325,
A329739,
A351202,
A383013,
A384889,
A384890.
-
Table[Length[Select[Subsets[Range[n]],UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]
-
lista(n)={my(o=(1-x^(n+1))/(1-x)*O(y^(n+2)),p=prod(i=1,n,1+o+x*y^(i+1)/(1-y),1/(1-y)));p=subst(serlaplace(p),x,1);Vec(p-1)} \\ Christian Sievers, Jun 18 2025
A384893
Triangle read by rows where T(n,k) is the number of subsets of {1..n} with k maximal anti-runs (increasing by more than 1).
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 7, 5, 2, 1, 1, 12, 10, 6, 2, 1, 1, 20, 20, 13, 7, 2, 1, 1, 33, 38, 29, 16, 8, 2, 1, 1, 54, 71, 60, 39, 19, 9, 2, 1, 1, 88, 130, 122, 86, 50, 22, 10, 2, 1, 1, 143, 235, 241, 187, 116, 62, 25, 11, 2, 1, 1, 232, 420, 468, 392, 267, 150, 75, 28, 12, 2, 1
Offset: 0
The subset {3,6,7,9,11,12} has maximal anti-runs ((3,6),(7,9,11),(12)), so is counted under T(12,3).
The subset {3,6,7,9,10,12} has maximal anti-runs ((3,6),(7,9),(10,12)), so is counted under T(12,3).
Row n = 5 counts the following subsets:
{} {1} {1,2} {1,2,3} {1,2,3,4} {1,2,3,4,5}
{2} {2,3} {2,3,4} {2,3,4,5}
{3} {3,4} {3,4,5}
{4} {4,5} {1,2,3,5}
{5} {1,2,4} {1,2,4,5}
{1,3} {1,2,5} {1,3,4,5}
{1,4} {1,3,4}
{1,5} {1,4,5}
{2,4} {2,3,5}
{2,5} {2,4,5}
{3,5}
{1,3,5}
Triangle begins:
1
1 1
1 2 1
1 4 2 1
1 7 5 2 1
1 12 10 6 2 1
1 20 20 13 7 2 1
1 33 38 29 16 8 2 1
1 54 71 60 39 19 9 2 1
1 88 130 122 86 50 22 10 2 1
1 143 235 241 187 116 62 25 11 2 1
1 232 420 468 392 267 150 75 28 12 2 1
1 376 744 894 806 588 363 188 89 31 13 2 1
For runs instead of anti-runs we have
A034839, for strict partitions
A116674.
For integer partitions instead of subsets we have
A268193, strict
A384905.
A384175 counts subsets with all distinct lengths of maximal runs, complement
A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts
A384878.
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Table[Length[Select[Subsets[Range[n]],Length[Split[#,#2!=#1+1&]]==k&]],{n,0,10},{k,0,n}]
A384177
Number of subsets of {1..n} with all distinct lengths of maximal anti-runs (increasing by more than 1).
Original entry on oeis.org
1, 2, 3, 5, 10, 19, 35, 62, 109, 197, 364, 677, 1251, 2288, 4143, 7443, 13318, 23837, 42809, 77216, 139751, 253293, 458800, 829237, 1494169, 2683316, 4804083, 8580293, 15301324, 27270061, 48607667, 86696300, 154758265, 276453311, 494050894, 882923051
Offset: 0
The subset {1,2,4,5,7,10} has maximal anti-runs ((1),(2,4),(5,7,10)), with lengths (1,2,3), so is counted under a(10).
The a(0) = 1 through a(5) = 19 subsets:
{} {} {} {} {} {}
{1} {1} {1} {1} {1}
{2} {2} {2} {2}
{3} {3} {3}
{1,3} {4} {4}
{1,3} {5}
{1,4} {1,3}
{2,4} {1,4}
{1,2,4} {1,5}
{1,3,4} {2,4}
{2,5}
{3,5}
{1,2,4}
{1,2,5}
{1,3,4}
{1,3,5}
{1,4,5}
{2,3,5}
{2,4,5}
These subsets are ranked by
A384879.
For equal instead of distinct lengths we have
A384889, for runs
A243815.
A034839 counts subsets by number of maximal runs, for strict partitions
A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement
A336866.
Cf.
A000009,
A010027,
A044813,
A047993,
A106529,
A123513,
A242882,
A325325,
A328592,
A329739,
A351202,
A384890.
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Table[Length[Select[Subsets[Range[n]],UnsameQ@@Length/@Split[#,#2!=#1+1&]&]],{n,0,10}]
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lista(n)={my(o=(1-x^(n+1))/(1-x)*O(y*y^n),p=prod(i=1,(n+1)\2,1+o+x*y^(2*i-1)/(1-y)^(i-1)));p=subst(serlaplace(p),x,1);Vec((p-y)/(1-y)^2)} \\ Christian Sievers, Jun 18 2025
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