A000295 -id:A000295 - cp's OEIS frontend

A001192 Number of full sets of size n.

1, 1, 1, 2, 9, 88, 1802, 75598, 6421599, 1097780312, 376516036188, 258683018091900, 355735062429124915, 978786413996934006272, 5387230452634185460127166, 59308424712939278997978128490, 1305926814154452720947815884466579

Offset: 0

N. J. A. Sloane

A set x is full if every element of x is also a subset of x.
Equals the subpartitions of Eulerian numbers (A000295(n)=2^n-n-1); see A115728 for the definition of subpartitions of a partition. - Paul D. Hanna, Jul 03 2006
Also number of transitive rooted identity trees with n branches. - Gus Wiseman, Dec 21 2016

			Examples of full sets are 0 := {}, 1 := {0}, 2 := {1,0}, 3a := {2,1,0}, 3b := { {1}, 1, 0}, 4a := { 3a, 2, 1, 0 }.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 123, Problem 20.
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).

T. D. Noe, Table of n, a(n) for n = 0..50
Bojan Bašić, Paul Ellis, Dana C. Ernst, Danijela Popović, and Nándor Sieben, Categories of impartial rulegraphs and gamegraphs, arXiv:2312.00650 [math.CO], 2023. See p. 20.
Alberto Casagrande, Carla Piazza, and Alberto Policriti, Is hyper-extensionality preservable under deletions of graph elements?, Elec. Notes Theor. Comp. Sci. (2016) Vol. 322, 103-118.
Richard Peddicord, The number of full sets with n elements, Proc. Amer. Math. Soc., 13 (1962), 825-828.
John Riordan, Letter to N. J. A. Sloane, Jul. 1970
Alexandru Ioan Tomescu, Sets as Graphs, Ph. D. Thesis, Università degli Studi di Udine, Dipartimento di Matematica e Informatica, Dottorato di Ricerca in Informatica, Dec. 2011.
Stephan Wagner, Asymptotic enumeration of extensional acyclic digraphs, in Proceedings of the SIAM Meeting on Analytic Algorithmics and Combinatorics (ANALCO12).
Gus Wiseman, Transitive rooted identity trees with n=5 branches

Cf. A000295, A004111, A115728, A182161, A279861, A279863.

A001192 := proc(n) option remember: if(n=0)then return 1: fi: return add((-1)^(n-k-1)*binomial(2^k-k,n-k)*procname(k), k=0..n-1); end: seq(A001192(n), n=0..16); # Nathaniel Johnston, Apr 18 2012

max = 16; f[x_] := Sum[a[n]*(x^n/(1+x)^2^n), {n, 0, max}] - 1; cc = CoefficientList[ Series[f[x], {x, 0, max}], x]; Table[a[n], {n, 0, max}] /. First[ Solve[ Thread[cc == 0]]] (* Jean-François Alcover, Nov 02 2011, after Vladeta Jovovic *)

{a(n)=polcoeff(x^n-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^(2^k-k-1) ), n)} \\ Paul D. Hanna, Jul 03 2006

1 = Sum_{n>=0} a(n)*x^n/(1+x)^(2^n). E.g., 1 = 1/(1+x) + 1*x/(1+x)^2 + 1*x^2/(1+x)^4 + 2*x^3/(1+x)^8 + 9*x^4/(1+x)^16 + 88*x^5/(1+x)^32 + 1802*x^6/(1+x)^64 + ... . - Vladeta Jovovic, May 26 2005
Equivalently, a(n) = (-1)^n*C(2^n+n-1, n) - Sum_{k=0..n-1} a(k)*(-1)^(n-k)*C(2^n+2^k+n-k-1, n-k). - Paul D. Hanna, May 26 2005
G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^(2^n-n-1) = 1*(1-x)^0 + 1*x*(1-x)^0 + 1*x^2*(1-x)^1 + 2*x^3*(1-x)^4 + 9*x^3*(1-x)^11 + ... + a(n)*x^n*(1-x)^(2^n-n-1) + ... . - Paul D. Hanna, Jul 03 2006

More terms from Ryan Propper, Jun 13 2005

A265901 Square array read by descending antidiagonals: A(n,1) = A188163(n), and for k > 1, A(n,k) = A087686(1+A(n,k-1)).

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 8, 15, 12, 6, 16, 31, 27, 14, 9, 32, 63, 58, 30, 21, 10, 64, 127, 121, 62, 48, 24, 11, 128, 255, 248, 126, 106, 54, 26, 13, 256, 511, 503, 254, 227, 116, 57, 29, 17, 512, 1023, 1014, 510, 475, 242, 120, 61, 38, 18, 1024, 2047, 2037, 1022, 978, 496, 247, 125, 86, 42, 19, 2048, 4095, 4084, 2046, 1992, 1006, 502, 253, 192, 96, 45, 20

Offset: 1

Antti Karttunen, Dec 18 2015

Square array read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
The topmost row (row 1) of the array is A000079 (powers of 2), and in general each row 2^k contains the sequence (2^n - k), starting from the term (2^(k+1) - k). This follows from the properties (3) and (4) of A004001 given on page 227 of Kubo & Vakil paper (page 3 in PDF).
Moreover, each row 2^k - 1 (for k >= 2) contains the sequence 2^n - n - (k-2), starting from the term (2^(k+1) - (2k-1)). To see why this holds, consider the definitions of sequences A162598 and A265332, the latter which also illustrates how the frequency counts Q_n for A004001 are recursively constructed (in the Kubo & Vakil paper).

			The top left corner of the array:
   1,  2,   4,   8,  16,   32,   64,  128,  256,   512,  1024, ...
   3,  7,  15,  31,  63,  127,  255,  511, 1023,  2047,  4095, ...
   5, 12,  27,  58, 121,  248,  503, 1014, 2037,  4084,  8179, ...
   6, 14,  30,  62, 126,  254,  510, 1022, 2046,  4094,  8190, ...
   9, 21,  48, 106, 227,  475,  978, 1992, 4029,  8113, 16292, ...
  10, 24,  54, 116, 242,  496, 1006, 2028, 4074,  8168, 16358, ...
  11, 26,  57, 120, 247,  502, 1013, 2036, 4083,  8178, 16369, ...
  13, 29,  61, 125, 253,  509, 1021, 2045, 4093,  8189, 16381, ...
  17, 38,  86, 192, 419,  894, 1872, 3864, 7893, 16006, 32298, ...
  18, 42,  96, 212, 454,  950, 1956, 3984, 8058, 16226, 32584, ...
  19, 45, 102, 222, 469,  971, 1984, 4020, 8103, 16281, 32650, ...
  20, 47, 105, 226, 474,  977, 1991, 4028, 8112, 16291, 32661, ...
  22, 51, 112, 237, 490,  999, 2020, 4065, 8158, 16347, 32728, ...
  23, 53, 115, 241, 495, 1005, 2027, 4073, 8167, 16357, 32739, ...
  25, 56, 119, 246, 501, 1012, 2035, 4082, 8177, 16368, 32751, ...
  28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, ...
  ...

Antti Karttunen, Table of n, a(n) for n = 1..210; the first 20 antidiagonals of array
T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
Index entries for Hofstadter-type sequences
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A267102.
Transpose: A265903.
Cf. A265900 (main diagonal).
Cf. A162598 (row index of n in array), A265332 (column index of n in array).
Cf. A004001, A051135, A088359, A087686.
Column 1: A188163.
Column 2: A266109.
Row 1: A000079 (2^n).
Row 2: A000225 (2^n - 1, from 3 onward).
Row 3: A000325 (2^n - n, from 5 onward).
Row 4: A000918 (2^n - 2, from 6 onward).
Row 5: A084634 (?, from 9 onward).
Row 6: A132732 (2^n - 2n + 2, from 10 onward).
Row 7: A000295 (2^n - n - 1, from 11 onward).
Row 8: A036563 (2^n - 3).
Row 9: A084635 (?, from 17 onward).
Row 12: A048492 (?, from 20 onward).
Row 13: A249453 (?, from 22 onward).
Row 14: A183155 (2^n - 2n + 1, from 23 onward. Cf. also A244331).
Row 15: A000247 (2^n - n - 2, from 25 onward).
Row 16: A028399 (2^n - 4).
Cf. also permutations A267111, A267112.

(define (A265901 n) (A265901bi (A002260 n) (A004736 n)))
(define (A265901bi row col) (if (= 1 col) (A188163 row) (A087686 (+ 1 (A265901bi row (- col 1))))))

For the first column k=1, A(n,1) = A188163(n), for columns k > 1, A(n,k) = A087686(1+A(n,k-1)).

A323819 Number of non-isomorphic connected set-systems covering n vertices.

Original entry on oeis.org

1, 1, 3, 30, 1912, 18662590, 12813206131799685, 33758171486592987138461432668177794, 1435913805026242504952006868879460423767388571975632398910903473535427583

Offset: 0

Gus Wiseman, Jan 30 2019

nonn

			Non-isomorphic representatives of the a(3) = 30 set-systems:
  {{1,2,3}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{2,3},{1,2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{3},{1,3},{2,3}}
  {{1},{2,3},{1,2,3}}
  {{3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}
  {{1},{3},{2,3},{1,2,3}}
  {{2},{3},{2,3},{1,2,3}}
  {{3},{1,2},{1,3},{2,3}}
  {{2},{1,3},{2,3},{1,2,3}}
  {{3},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,3},{2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{2,3},{1,2,3}}
  {{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{1,3},{2,3},{1,2,3}}
  {{2},{3},{1,3},{2,3},{1,2,3}}
  {{3},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3},{1,2,3}}
  {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Alois P. Heinz, Table of n, a(n) for n = 0..12
Thomas Delacroix, Meaningful objective frequency-based interesting pattern mining, Thesis, 2021.
Geon Lee, Seokbum Yoon, Jihoon Ko, Hyunju Kim, and Kijung Shin, Hypergraph Motifs and Their Extensions Beyond Binary, arXiv:2310.15668 [cs.SI], 2023.

Cf. A000295, A003465, A016031, A048143, A055621 (not necessarily connected), A293510, A317795, A323817, A323818 (labeled case).

nmax = 12;
b[n_, i_, l_] := b[n, i, l] = If[n == 0, 2^Function[w, Sum[Product[2^GCD[t, l[[h]]], {h, 1, Length[l]}], {t, 1, w}]/w][If[l == {}, 1, LCM @@ l]], If[i < 1, 0, Sum[b[n - i*j, i - 1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]]];
f[n_] := If[n == 0, 2, b[n, n, {}] - b[n - 1, n - 1, {}]]/2;
A055621 = f /@ Range[0, nmax];
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
Join[{1}, EULERi[A055621 // Rest]] (* Jean-François Alcover, Jan 31 2020, after Alois P. Heinz in A055621 *)

Inverse Euler transform of A055621.

A014473 Pascal's triangle - 1: Triangle read by rows: T(n, k) = A007318(n, k) - 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 5, 3, 0, 0, 4, 9, 9, 4, 0, 0, 5, 14, 19, 14, 5, 0, 0, 6, 20, 34, 34, 20, 6, 0, 0, 7, 27, 55, 69, 55, 27, 7, 0, 0, 8, 35, 83, 125, 125, 83, 35, 8, 0, 0, 9, 44, 119, 209, 251, 209, 119, 44, 9, 0, 0, 10, 54, 164, 329, 461, 461, 329, 164, 54, 10, 0

Offset: 0

N. J. A. Sloane

Indexed as a square array A(n,k): If X is an (n+k)-set and Y a fixed k-subset of X then A(n,k) is equal to the number of n-subsets of X intersecting Y. - Peter Luschny, Apr 20 2012

			Triangle begins:
   0;
   0, 0;
   0, 1,  0;
   0, 2,  2,  0;
   0, 3,  5,  3,  0;
   0, 4,  9,  9,  4,  0;
   0, 5, 14, 19, 14,  5, 0;
   0, 6, 20, 34, 34, 20, 6, 0;
   ...
Seen as a square array read by antidiagonals:
  [0] 0, 0,  0,  0,   0,   0,   0,    0,    0,    0,    0,     0, ... A000004
  [1] 0, 1,  2,  3,   4,   5,   6,    7,    8,    9,   10,    11, ... A001477
  [2] 0, 2,  5,  9,  14,  20,  27,   35,   44,   54,   65,    77, ... A000096
  [3] 0, 3,  9, 19,  34,  55,  83,  119,  164,  219,  285,   363, ... A062748
  [4] 0, 4, 14, 34,  69, 125, 209,  329,  494,  714, 1000,  1364, ... A063258
  [5] 0, 5, 20, 55, 125, 251, 461,  791, 1286, 2001, 3002,  4367, ... A062988
  [6] 0, 6, 27, 83, 209, 461, 923, 1715, 3002, 5004, 8007, 12375, ... A124089

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened
Milan Janjic, Two Enumerative Functions

Triangle without zeros: A014430.
Related: A323211 (A007318(n, k) + 1).
A000295 (row sums), A059841 (alternating row sums), A030662(n-1) (central terms).
Columns include A000096, A062748, A062988, A063258.
Diagonals of A(n, n+d): A030662 (d=0), A010763 (d=1), A322938 (d=2).
Cf. A000004, A001477, A007318, A030662, A059841, A109128, A124089, A129696.

a014473 n k = a014473_tabl !! n !! k
a014473_row n = a014473_tabl !! n
a014473_tabl = map (map (subtract 1)) a007318_tabl
-- Reinhard Zumkeller, Apr 10 2012

[Binomial(n,k)-1: k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 08 2024

with(combstruct): for n from 0 to 11 do seq(-1+count(Combination(n), size=m), m = 0 .. n) od; # Zerinvary Lajos, Apr 09 2008
# The rows of the square array:
Arow := proc(n, len) local gf, ser;
gf := (x - 1)^(-n - 1) + (-1)^(n + 1)/(x*(x - 1));
ser := series(gf, x, len+2): seq((-1)^(n+1)*coeff(ser, x, j), j=0..len) end:
for n from 0 to 9 do lprint([n], Arow(n, 12)) od; # Peter Luschny, Feb 13 2019

Table[Binomial[n,k] -1, {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 08 2024 *)

flatten([[binomial(n,k)-1 for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 08 2024

G.f.: x^2*y/((1 - x)*(1 - x*y)*(1 - x*(1 + y))). - Ralf Stephan, Jan 24 2005
T(n,k) = A109128(n,k) - A007318(n,k), 0 <= k <= n. - Reinhard Zumkeller, Apr 10 2012
T(n, k) = T(n-1, k-1) + T(n-1, k) + 1, 0 < k < n with T(n, 0) = T(n, n) = 0. - Reinhard Zumkeller, Jul 18 2015
If seen as a square array read by antidiagonals the generating function of row n is: G(n) = (x - 1)^(-n - 1) + (-1)^(n + 1)/(x*(x - 1)). - Peter Luschny, Feb 13 2019
From G. C. Greubel, Apr 08 2024: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..floor(n/2)} T(n-k, k) = A129696(n-2).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = b(n-1), where b(n) is the repeating pattern {0, 0, -1, -2, -1, 1, 1, -1, -2, -1, 0, 0}_{n=0..11}, with b(n) = b(n-12). (End)

More terms from Erich Friedman

A066810 Expansion of x^2/((1-3x)(1-2*x)^2).

Original entry on oeis.org

0, 0, 1, 7, 33, 131, 473, 1611, 5281, 16867, 52905, 163835, 502769, 1532883, 4651897, 14070379, 42456897, 127894979, 384799049, 1156756443, 3475250065, 10436235955, 31330727961, 94038321227, 282211432673, 846835624611, 2540926304233, 7623651327931, 22872765923121

Offset: 0

N. J. A. Sloane, Jan 25 2002

Binomial transform of A000295.
a(n) = A112626(n, 2). - Ross La Haye, Jan 11 2006
Let Q be a binary relation on the power set P(A) of a set A having n = |A| elements such that for all x,y of P(A), xQy if x is a proper subset of y and |y| - |x| > 1. Then a(n) = |Q|. - Ross La Haye, Jan 11 2008
a(n) is the number of n-digit ternary sequences that have at least two 0's. - Geoffrey Critzer, Apr 14 2009

Harry J. Smith, Table of n, a(n) for n = 0..200
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Index entries for linear recurrences with constant coefficients, signature (7,-16,12).

Column k=1 of A238858 (with different offset).

List([0..30], n-> 3^n - 2^n - n*2^(n-1)); # G. C. Greubel, Nov 18 2019

[3^n-2^n-n*2^(n-1): n in [0..30]]; // Vincenzo Librandi, Nov 29 2015

seq(3^n - 2^n - n*2^(n-1), n=0..30); # G. C. Greubel, Nov 18 2019

RecurrenceTable[{a[n]==3*a[n-1] + (n-1) 2^(n-2), a[0]==0}, a, {n, 0, 30}] (* Geoffrey Critzer, Apr 14 2009 *)
CoefficientList[Series[x^2/((1-3x)(1-2x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Nov 29 2015 *)

a(n) = 3^n -2^n -n*2^(n-1) \\ Harry J. Smith, Mar 29 2010

[3^n - 2^n - n*2^(n-1) for n in (0..30)] # G. C. Greubel, Nov 18 2019

a(n) = 3^n - 2^n - n*2^(n-1).
From Ross La Haye, Apr 26 2006: (Start)
a(n) = A000244(n) - A001792(n).
a(n) = Sum_{k=2..n} binomial(n,k)2^(n-k). (End)
Inverse binomial transform of A086443. - Ross La Haye, Apr 29 2006
Convolution of A000244 beginning [0,1,3,9,27,81,...] and A001787. - Ross La Haye, Feb 15 2007
From Geoffrey Critzer, Apr 14 2009: (Start)
E.g.f.: exp(2*x)*(exp(x) - x - 1).
a(n) = 3*a(n-1) + (n-1)*2^(n-2). (End)

Additional comments from Ross La Haye, Sep 27 2005

A112493 Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 11, 25, 15, 1, 26, 130, 210, 105, 1, 57, 546, 1750, 2205, 945, 1, 120, 2037, 11368, 26775, 27720, 10395, 1, 247, 7071, 63805, 247555, 460845, 405405, 135135, 1, 502, 23436, 325930, 1939630, 5735730, 8828820, 6756750, 2027025, 1

Offset: 0

Wolfdieter Lang, Oct 14 2005

Previous name was: Coefficient triangle of polynomials used for e.g.f.s of Stirling2 diagonals.
For the o.g.f. of diagonal k of the Stirling2 triangle one has a similar result. See A008517 (second-order Eulerian triangle).
A(m,x), the o.g.f. for column m, satisfies the recurrence A(m,x) = x*(x*(d/dx)A(m-1,x) + m*A(m-1,x))/(1-(m+1)*x), for m >= 1 and A(0,x) = 1/(1-x).
The e.g.f. for the sequence in column k+1, k >= 0, of A008278, i.e., for the diagonal k >= 0 of the Stirling2 triangle A048993, is exp(x)*Sum_{m=0..k} a(k,m)*(x^(m+k))/(m+k)!.
It appears that the triangles in this sequence and A124324 have identical columns, except for shifts. - Jörgen Backelin, Jun 20 2022
A refined version of this triangle is given in A356145, which contains a link providing the precise relationship between A124324 and this entry, confirming Jörgen Backelin's observation above. - Tom Copeland, Sep 24 2022

			Triangle starts:
  [1]
  [1, 1]
  [1, 4,  3]
  [1, 11, 25,  15]
  [1, 26, 130, 210,  105]
  [1, 57, 546, 1750, 2205, 945]
  ...
The e.g.f. of [0,0,1,7,25,65,...], the k=3 column of A008278, but with offset n=0, is exp(x)*(1*(x^2)/2! + 4*(x^3)/3! + 3*(x^4)/4!).
Third row [1,4,3]: There are three plane increasing trees on 3 vertices. The number of colors are shown to the right of a vertex.
...................................................
....1o.(1+t)...........1o.t*(1+t).....1o.t*(1+t)...
....|................. /.\............/.\..........
....|................ /...\........../...\.........
....2o.(1+t)........2o.....3o......3o....2o........
....|..............................................
....|..............................................
....3o.............................................
...................................................
The total number of trees is (1+t)^2 + t*(1+t) + t*(1+t) = 1+4*t+3*t^2 = R(2,t).

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA], 2005.
Wolfdieter Lang, First ten rows.
MathOverflow, Recursion for row polynomials of A112493, (2025).
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.

Row sums give A006351(k+1), k>=0.
The column sequences start with A000012 (powers of 1), A000295 (Eulerian numbers), A112495, A112496, A112497.
Cf. A008278, A008517, A048993, A054589, A075856, A134991, A201637, A124324.
Antidiagonal sums give A000110.
Cf. A356145.

T := (n, k) -> add(combinat:-eulerian2(n, j)*binomial(n-j, n-k), j=0..n):
seq(seq(T(n, k), k=0..n), n=0..9); # Peter Luschny, Apr 11 2016

max = 11; f[x_, t_] := -1 - (1 + t)/t*ProductLog[-t/(1 + t)*Exp[(x - t)/(1 + t)]]; coes = CoefficientList[ Series[f[x, t], {x, 0, max}, {t, 0, max}], {x, t}]* Range[0, max]!; Table[coes[[n, k]], {n, 0, max}, {k, 1, n - 1}] // Flatten (* Jean-François Alcover, Nov 22 2012, from e.g.f. *)

a(k, m) = 0 if k < m, a(k, -1):=0, a(0, 0)=1, a(k, m)=(m+1)*a(k-1, m) + (k+m-1)*a(k-1, m-1) else.
From Peter Bala, Sep 30 2011: (Start)
E.g.f.: A(x,t) = -1-((1+t)/t)*LambertW(-(t/(1+t))*exp((x-t)/(1+t))) = x + (1+t)*x^2/2! + (1+4*t+3*t^2)*x^3/3! + .... A(x,t) is the inverse function of (1+t)*log(1+x)-t*x.
A(x,t) satisfies the partial differential equation (1-x*t)*dA/dx = 1 + A + t*(1+t)*dA/dt. It follows that the row generating polynomials R(n,t) satisfy the recurrence R(n+1,t) =(n*t+1)*R(n,t) + t*(1+t)*dR(n,t)/dt. Cf. A054589 and A075856. The polynomials t/(1+t)*R(n,t) are the row polynomials of A134991.
The generating function A(x,t) satisfies the autonomous differential equation dA/dx = (1+A)/(1-t*A). Applying [Bergeron et al., Theorem 1] gives a combinatorial interpretation for the row generating polynomials R(n,t): R(n,t) counts plane increasing trees on n+1 vertices where the non-leaf vertices of outdegree k come in t^(k-1)*(1+t) colors. An example is given below. Cf. A006351, which corresponds to the case t = 1. Applying [Dominici, Theorem 4.1] gives the following method for calculating the row polynomials R(n,t): Let f(x) = (1+x)/(1-x*t). Then R(n,t) = (f(x)*d/dx)^n(f(x)) evaluated at x = 0. (End)
Sum_{j=0..n} T(n-j,j) = A000110(n). - Alois P. Heinz, Jun 20 2022
From Mikhail Kurkov, Apr 01 2025: (Start)
E.g.f.: B(y) = -w/(x*(1+w)) where w = LambertW(-x/(1+x)*exp((y-x)/(1+x))) satisfies the first-order ordinary differential equation (1+x)*B'(y) = B(y)*(1+x*B(y))^2, hence row polynomials are P(n,x) = P(n-1,x) + x*Sum_{j=0..n-1} binomial(n, j)*P(j,x)*P(n-j-1,x) for n > 0 with P(0,x) = 1 (see MathOverflow link).
Conjecture: row polynomials are P(n,x) = Sum_{i=0..n} Sum_{j=0..i} Sum_{k=0..j} (n+i)!*Stirling1(n+j-k,j-k)*x^k*(x+1)^(j-k)*(-1)^(j+k)/((n+j-k)!*(i-j)!*k!). (End)
Conjecture: g.f. satisfies 1/(1 - x - x*y/(1 - 2*x - 2*x*y/(1 - 3*x - 3*x*y/(1 - 4*x - 4*x*y/(1 - 5*x - 5*x*y/(1 - ...)))))) (see A383019 for conjectures about combinatorial interpretation and algorithm for efficient computing). - Mikhail Kurkov, Apr 21 2025

New name from Peter Luschny, Apr 11 2016

A122753 Triangle of coefficients of (1 - x)^n*B_n(x/(1 - x)), where B_n(x) is the n-th Bell polynomial.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 1, -1, 0, 1, 4, -5, 1, 0, 1, 11, -14, 1, 2, 0, 1, 26, -24, -29, 36, -9, 0, 1, 57, 1, -244, 281, -104, 9, 0, 1, 120, 225, -1259, 1401, -454, -83, 50, 0, 1, 247, 1268, -5081, 4621, 911, -3422, 1723, -267, 0, 1, 502, 5278, -16981, 5335, 30871

Offset: 0

Roger L. Bagula and Gary W. Adamson, Sep 21 2006

The n-th row consists of the coefficients in the expansion of Sum_{j=0..n} A048993(n,j)*x^j*(1 - x)^(n - j), where A048993 is the triangle of Stirling numbers of second kind.

			Triangle begins:
    1;
    0, 1;
    0, 1;
    0, 1,   1,   -1;
    0, 1,   4,   -5,     1;
    0, 1,  11,  -14,     1,    2;
    0, 1,  26,  -24,   -29,   36,   -9;
    0, 1,  57,    1,  -244,  281, -104,     9;
    0, 1, 120,  225, -1259, 1401, -454,   -83,   50;
    0, 1, 247, 1268, -5081, 4621,  911, -3422, 1723, -267;
    ... reformatted and extended. - _Franck Maminirina Ramaharo_, Oct 10 2018

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 824-825.

G. C. Greubel, Rows n = 0..50 of the irregular triangle, flattened
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].
Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Eric Weisstein's World of Mathematics, Bell Polynomial
Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind

Cf. A000110, A048993, A088996, A122610.

Cf. A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221.

Table[CoefficientList[Sum[StirlingS2[m, n]*x^n*(1-x)^(m-n), {n,0,m}], x], {m,0,10}]//Flatten

P(x, n) := expand(sum(stirling2(n, j)*x^j*(1 - x)^(n - j), j, 0, n))$
T(n, k) := ratcoef(P(x, n), x, k)$
tabf(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, hipow(P(x, n), x)))$ /* Franck Maminirina Ramaharo, Oct 10 2018 */

def p(n,x): return sum( stirling_number2(n, j)*x^j*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
flatten([T(n) for n in (0..12)]) # G. C. Greubel, Jul 15 2021

From Franck Maminirina Ramaharo, Oct 10 2018: (Start)
E.g.f.: exp((x/(1 - x))*(exp((1 - x)*y) - 1)).
T(n,1) = A000295(n-1). (End)

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 10 2018

A143291 Triangle T(n,k), n>=2, 0<=k<=n-2, read by rows: numbers of binary words of length n containing at least one subword 10^{k}1 and no subwords 10^{i}1 with i

Original entry on oeis.org

1, 3, 1, 8, 2, 1, 19, 4, 2, 1, 43, 8, 3, 2, 1, 94, 15, 5, 3, 2, 1, 201, 27, 9, 4, 3, 2, 1, 423, 48, 15, 6, 4, 3, 2, 1, 880, 84, 24, 10, 5, 4, 3, 2, 1, 1815, 145, 38, 16, 7, 5, 4, 3, 2, 1, 3719, 248, 60, 24, 11, 6, 5, 4, 3, 2, 1, 7582, 421, 94, 35, 17, 8, 6, 5, 4, 3, 2, 1, 15397, 710, 146, 51, 25, 12, 7, 6, 5, 4, 3, 2, 1

Offset: 2

Alois P. Heinz, Aug 04 2008

T(n,k) = number of subset S of {1,2,...,n+1} such that |S| > 1 and min(S*) = k, where S* is the set {x(2)-x(1), x(3)-x(2), ..., x(h+1)-x(h)} when the elements of S are written as x(1) < x(2) < ... < x(h+1); if max(S*) is used in place of min(S*), the result is the array at A255874. - Clark Kimberling, Mar 08 2015

			T (5,1) = 4, because there are 4 words of length 5 containing at least one subword 101 and no subword 11: 00101, 01010, 10100, 10101.
Triangle begins:
    1;
    3,  1;
    8,  2,  1;
   19,  4,  2, 1;
   43,  8,  3, 2, 1;
   94, 15,  5, 3, 2, 1;
  201, 27,  9, 4, 3, 2, 1;
  423, 48, 15, 6, 4, 3, 2, 1;

Alois P. Heinz, Rows n = 2..142, flattened

Columns k=0-10 give: A008466, A143281, A143282, A143283, A143284, A143285, A143286, A143287, A143288, A143289, A143290.
Row sums are in A000295.
Cf. A141539.

R:= PowerSeriesRing(Integers(), 50);
A143291:= func< n,k | Coefficient(R!( x^k/((x^(k-1) +x-1)*(x^k +x-1)) ), n) >;
[A143291(n,k): k in [2..n], n in [2..12]]; // G. C. Greubel, Jun 01 2025

as:= proc (n, k) option remember;
       if k=0 then 2^n
     elif n<=k and n>=0 then n+1
     elif n>0 then as(n-1, k) +as(n-k-1, k)
     else as(n+1+k, k) -as(n+k, k)
       fi
     end:
T:= (n, k)-> as(n, k) -as(n, k+1):
seq(seq(T(n, k), k=0..n-2), n=2..15);

as[n_, k_] := as[n, k] = Which[ k == 0, 2^n, n <= k && n >= 0, n+1, n > 0, as[n-1, k] + as[n-k-1, k], True, as[n+1+k, k] - as[n+k, k] ]; t [n_, k_] := as[n, k] - as[n, k+1]; Table[Table[t[n, k], {k, 0, n-2}], {n, 2, 14}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)

@CachedFunction
def b(n,k):
    if k==0: return 2^n
    elif n <= k and n>=0: return n+1
    elif n>0: return b(n-1,k) + b(n-k-1,k)
    else: return b(n+k+1,k) - b(n+k,k)
def A143291(n,k): return b(n,k) - b(n,k+1)
print(flatten([[A143291(n,k) for k in range(n-1)] for n in range(2,16)])) # G. C. Greubel, Jun 01 2025

G.f. of column k: x^(k+2) / ((x^(k+1)+x-1)*(x^(k+2)+x-1)).

A187059 The exponent of highest power of 2 dividing the product of the elements of the n-th row of Pascal's triangle (A001142).

Original entry on oeis.org

0, 0, 1, 0, 5, 2, 4, 0, 17, 10, 12, 4, 18, 8, 11, 0, 49, 34, 36, 20, 42, 24, 27, 8, 58, 36, 39, 16, 47, 22, 26, 0, 129, 98, 100, 68, 106, 72, 75, 40, 122, 84, 87, 48, 95, 54, 58, 16, 162, 116, 119, 72, 127, 78, 82, 32, 147, 94, 98, 44, 108, 52, 57, 0, 321, 258, 260, 196, 266, 200, 203, 136, 282, 212, 215, 144, 223, 150, 154, 80, 322, 244, 247, 168, 255, 174, 178, 96, 275, 190, 194, 108, 204, 116, 121, 32, 418, 324, 327, 232, 335

Offset: 0

Bruce Reznick, Mar 05 2011

The exponent of the highest power of 2 which divides Product_{k=0..n} binomial(n, k). This can be computed using de Polignac's formula.

This is the function ord_2(Ḡ_n) extensively studied in Lagarias-Mehta (2014), and plotted in Fig. 1.1. - Antti Karttunen, Oct 22 2014

			For example, if n = 4, the power of 2 that divides 1*4*6*4*1 is 5.

I. Niven, H. S. Zuckerman, H. L. Montgomery, An Introduction to the Theory of Numbers, Wiley, 1991, pages 182, 183, 187 (Ex. 34).

Antti Karttunen and Paul Tek, Table of n, a(n) for n = 0..8191 (First 4096 terms from Karttunen)
J. Lagarias, Products of binomial coefficients and Farey fractions, Lecture in DIMACS Conference on Challenges of Identifying Integer Sequences, October 9, 2014; Part 1, Part 2.
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.

Row sums of triangular table A065040.
Row 1 of array A249421.
Cf. A000295 (a(2^k-2)), A000337 (a(2^k)), A005803 (a(2^k-3)), A036799 (a(2^k+1)), A109363 (a(2^k-4)).
Cf. A000178, A000788, A001142, A002109, A007318, A007814, A174605, A249152, A249150, A249151, A249154, A249343, A249345, A249346, A249347.

a187059 = a007814 . a001142  -- Reinhard Zumkeller, Mar 16 2015

a[n_] := Sum[IntegerExponent[Binomial[n, k], 2], {k, 0, n}]; Array[a, 100, 0]
A187059[n_] := Sum[#*((#+1)*2^k - n - 1) & [Floor[n/2^k]], {k, Floor[Log2[n]]}];
Array[A187059, 100, 0] (* Paolo Xausa, Feb 11 2025 *)
2*Accumulate[#] - Range[Length[#]]*# & [DigitCount[Range[0, 99], 2, 1]] (* Paolo Xausa, Feb 11 2025 *)

a(n)=sum(k=0,n,valuation(binomial(n,k),2))

\\ Much faster version, based on code for A065040 by Charles R Greathouse IV which if reduced even further gives the formula a(n) = 2*A000788(n) - A249154(n):
A065040(m,k) = (hammingweight(k)+hammingweight(m-k)-hammingweight(m));
A187059(n) = sum(k=0, n, A065040(n, k));
for(n=0, 4095, write("b187059.txt", n, " ", A187059(n)));
\\ Antti Karttunen, Oct 25 2014

def A187059(n): return (n+1)*n.bit_count()+sum((m:=1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1)) # Chai Wah Wu, Nov 11 2024

a(2^k-1) = 0 (19th century); a(2^k) = (k-1)*2^k+1 for k >= 1. (Use de Polignac.)
a(n) = Sum_{i=0..n} A065040(n,i) [where the entries of triangular table A065040(m,k) give the exponent of the maximal power of 2 dividing binomial coefficient A007318(m,k)].
a(n) = A007814(A001142(n)). - Jason Kimberley, Nov 02 2011
a(n) = A249152(n) - A174605(n). [Exponent of 2 in the n-th hyperfactorial minus exponent of 2 in the n-th superfactorial. Cf. for example Lagarias & Mehta paper or Peter Luschny's formula for A001142.] - Antti Karttunen, Oct 25 2014
a(n) = 2*A000788(n) - A249154(n). - Antti Karttunen, Nov 02 2014
a(n) = Sum_{i=1..n} (2*i-n-1)*v_2(i), where v_2(i) = A007814(i) is the exponent of the highest power of 2 dividing i. - Ridouane Oudra, Jun 02 2022
a(n) = Sum_{k=1..floor(log_2(n))} t*((t+1)*2^k - n - 1), where t = floor(n/(2^k)). - Paolo Xausa, Feb 11 2025, derived from Ridouane Oudra's formula above.

Name clarified by Antti Karttunen, Oct 22 2014

A251268 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with no 2X2 subblock having x11-x00 less than x10-x01.

Original entry on oeis.org

11, 26, 35, 57, 114, 108, 120, 313, 480, 337, 247, 772, 1667, 2058, 1049, 502, 1775, 4930, 9109, 8812, 3268, 1013, 3894, 13052, 32636, 49872, 37772, 10179, 2036, 8277, 31936, 100843, 217634, 273607, 161906, 31707, 4083, 17224, 73805, 279718, 790734

Offset: 1

R. H. Hardin, Dec 01 2014

Table starts
.....11.......26........57........120.........247..........502.........1013
.....35......114.......313........772........1775.........3894.........8277
....108......480......1667.......4930.......13052........31936........73805
....337.....2058......9109......32636......100843.......279718.......715685
...1049.....8812.....49872.....217634......790734......2510004......7189937
...3268....37772....273607....1457326.....6247708.....22806904.....73607411
..10179...161906...1501739....9772880....49523566....208452452....760734085
..31707...694042...8244503...65582500...393172015...1910905110...7901650053
..98764..2975162..45265163..440223510..3123669457..17543333688..82288916360
.307641.12753740.248529844.2955392154.24825649060.161181383956.858174176431

			Some solutions for n=4 k=4
..0..0..0..0..0....0..1..1..1..1....0..0..0..0..1....0..0..1..0..1
..1..1..1..1..1....0..0..0..1..1....0..1..1..1..1....0..0..0..1..1
..1..1..1..1..1....1..1..1..1..1....0..0..0..0..0....0..1..1..1..1
..1..1..1..1..1....0..0..0..0..0....0..0..1..1..1....0..0..0..0..0
..0..0..0..0..1....0..0..1..1..1....0..1..0..1..1....0..0..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..479

Column 1 is A052550(n+2)

Row 1 is A000295(n+3)

Empirical for column k:
k=1: a(n) = 3*a(n-1) +a(n-2) -2*a(n-3)
k=2: a(n) = 5*a(n-1) -2*a(n-2) -5*a(n-3) +2*a(n-4)
k=3: [order 10]
k=4: [order 16]
k=5: [order 36]
k=6: [order 62]
Empirical for row n:
n=1: a(n) = 4*a(n-1) -5*a(n-2) +2*a(n-3)
n=2: a(n) = 6*a(n-1) -14*a(n-2) +16*a(n-3) -9*a(n-4) +2*a(n-5)
n=3: [order 8]
n=4: [order 10]
n=5: [order 12]
n=6: [order 14]
n=7: [order 16]

cp's OEIS Frontend

A001192 Number of full sets of size n.

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A265901 Square array read by descending antidiagonals: A(n,1) = A188163(n), and for k > 1, A(n,k) = A087686(1+A(n,k-1)).

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A323819 Number of non-isomorphic connected set-systems covering n vertices.

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A014473 Pascal's triangle - 1: Triangle read by rows: T(n, k) = A007318(n, k) - 1.

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A066810 Expansion of x^2/((1-3*x)*(1-2*x)^2).

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A112493 Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.

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A066810 Expansion of x^2/((1-3x)(1-2*x)^2).