A006116 -id:A006116 - cp's OEIS frontend

A015196 Sum of Gaussian binomial coefficients for q=10.

1, 2, 13, 224, 13435, 2266646, 1348019857, 2269339773068, 13484735901526279, 226960944509263279490, 13485189809930561625032701, 2269636415245291711513986785912, 1348523520252401463276762566348539123

Offset: 0

N. J. A. Sloane, Olivier Gérard

nonn

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Vincenzo Librandi, Table of n, a(n) for n = 0..60
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A006116, A006117, A006118, A006119, A006120, A006121, A006122, A015195.

Row sums of triangle A022174.

Total/@Table[QBinomial[n, m, 10], {n, 0, 20}, {m, 0, n}] (* Vincenzo Librandi, Nov 01 2012 *)
Flatten[{1,RecurrenceTable[{a[n]==2*a[n-1]+(10^(n-1)-1)*a[n-2],a[0]==1,a[1]==2},a,{n,1,15}]}] (* Vaclav Kotesovec, Aug 21 2013 *)

a(n) = 2*a(n-1)+(10^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - Vaclav Kotesovec, Aug 21 2013

a(n) ~ c * 10^(n^2/4), where c = EllipticTheta[3,0,1/10]/QPochhammer[1/10,1/10] = 1.348524024616... if n is even and c = EllipticTheta[2,0,1/10]/QPochhammer[1/10,1/10] = 1.2763120346269... if n is odd. - Vaclav Kotesovec, Aug 21 2013

A198260 Runs of 1s in binary strings corresponding to subgroups of nimber addition.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 2, 2, 4, 3, 2, 3, 4, 2, 1, 2, 2, 2, 2, 4, 3, 2, 4, 4, 2, 2, 2, 4, 4, 4, 3, 2, 4, 4, 4, 4, 2, 4, 3, 2, 4, 4, 4, 4, 4, 8, 7, 3, 5, 6, 2, 3, 4, 4, 4, 4, 6, 5, 4, 7, 8, 2, 3, 4, 2, 1, 2, 2, 2, 2, 4, 3, 2, 4, 4, 2, 2, 2, 4, 4, 4, 3, 2, 4, 4, 4, 4, 2, 4, 3, 2, 4, 4

Offset: 0

Tilman Piesk, Oct 22 2011

nonn

Counts the runs of consecutive ones in the binary representation of A190939. All positive integers appear in this sequence.

The first A006116(n) entries contain all integers from 1 to 2^(n-1). E.g. the first 67 entries contain all integers from 1 to 8. How often each integer appears is counted by the triangle A227961.

Tilman Piesk, Table of n, a(n) for n = 0..2824

Cf. A190939 (subgroups of nimber addition interpreted as binary numbers).
Cf. A069010 (runs of 1s).
Cf. A227961 (corresponding triangle).

a(n) = A069010(A190939(n)).

Changed offset to 0 as in A190939 by Tilman Piesk, Jan 25 2012

Significant edit by Tilman Piesk, Aug 01 2013

A227960 Big equivalence classes (A227723) related to subgroups of nimber addition (A190939).

Original entry on oeis.org

1, 3, 6, 15, 24, 60, 105, 255, 384, 960, 1632, 1680, 4080, 15555, 27030, 65535, 98304, 245760, 417792, 430080, 1044480, 1582080, 3947520, 3982080, 6908160, 6919680, 16776960, 106991625, 267448335, 1019462460, 1771476585, 4294967295

Offset: 0

Tilman Piesk, Aug 01 2013

A subsequence of A227723, showing all the big equivalence classes that contain Boolean functions related to subgroups of nimber addition (A190939).
Forms a triangle with row lengths A034343 = 1, 1, 2, 4, 8, 16, 36, 80...:
1,
3,
6, 15,
24, 60, 105, 255,
384, 960, 1632, 1680, 4080, 15555, 27030, 65535...
The left column a( 1,2,4,8,16,32,68,148... ) = a( A076766 ) = 3 ,6, 24, 384, 98304... is probably A001146 * 3/2, which is also A006017( A000079 ).
The first A076766(n) entries correspond to the first A006116(n) entries of A190939. (The first 148 here, for n = 7,  correspond to the first 29212 there.) The entries of A190939 can be generated from this sequence.
Among the first A076766(n) entries are A076831(n;0...n) with weight 2^0...2^n. (Among the first 148 are 1, 7, 23, 43, 43, 23, 7, 1 with weights 1, 2, 4, 8, 16, 32, 64, 128.)
a(n) appears to be divisible by 3 for n>0, and the odd part of a(n) is almost always squarefree. - Ralf Stephan, Aug 02 2013

Tilman Piesk, Table of n, a(n) for n = 0..147
Tilman Piesk, a(n) for n = 0..147 and corresponding entries of A190939 in reverse binary. Prime factors and numbers of prime factors (A001222).
Tilman Piesk, Subgroups of nimber addition (Wikiversity)

Subsequence of A227723 (all becs). All entries are also in A227963 (all sona-secs). Neither shares the property of divisibility by 3.
A190939, A034343, A076766, A076831.
The prime factors contain many prime factors of Fermat numbers (A023394).

a( A076766 - 1 ) = A001146 - 1 = A051179.

a( A076766 ) = A001146 * 3/2 (probably).

A228465 Recurrence a(n) = a(n-1) + 2^n*a(n-2) with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 9, 25, 313, 1913, 41977, 531705, 22023929, 566489849, 45671496441, 2366013917945, 376506912762617, 39141278944373497, 12376519796349807353, 2577539376694811306745, 1624792742123856760679161, 677311275106408471956040441, 852536648457739021814912002809

Offset: 0

Vaclav Kotesovec, Aug 22 2013

nonn

Generally (if p>0, q>1), recurrence a(n) = b*a(n-1) + (p*q^n+d)*a(n-2), a(n) is asymptotic to c*q^(n^2/4)*(p*q)^(n/2), where c is for fixed parameters b, p, d, q, a(0), a(1) constant, independent on n.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A006116, A006117, A006118, A006119, A006120, A006121, A006122.

[n le 2 select (n-1) else Self(n-1)+Self(n-2)*2^(n-1): n in [1..20]]; // Vincenzo Librandi, Aug 23 2013

RecurrenceTable[{a[n]==a[n-1]+2^n*a[n-2],a[0]==0,a[1]==1},a,{n,0,20}]
(* Alternative: *)
a[n_] := Sum[2^(k^2-1) QBinomial[n - k , k - 1, 2], {k, 1, n}];
Table[a[n], {n, 0, 19}] (* After Vladimir Kruchinin. Peter Luschny, Jan 20 2020 *)

def a(n):
    return sum(2^(k^2 - 1)*q_binomial(n-k , k-1, 2) for k in (1..n))
print([a(n) for n in range(20)]) # Peter Luschny, Jan 20 2020

a(n) ~ c * 2^(n^2/4 + n/2), where c = 0.548441579870783378573455400152590154... if n is even and c = 0.800417244834941368929416800341853541... if n is odd.

a(n) = Sum_{k=1..floor(n/2+1/2)} qbinomial(n-k,k-1)*2^(k^2-1), where q-binomial is triangle A022166, that is, with q=2. - Vladimir Kruchinin, Jan 20 2020

A015197 Sum of Gaussian binomial coefficients for q=11.

Original entry on oeis.org

1, 2, 14, 268, 19156, 3961832, 3092997464, 7024809092848, 60287817008722576, 1505950784990730735392, 142158530752430089391520224, 39060769254069395008311334483648, 40559566021977397260316290099710383936

Offset: 0

N. J. A. Sloane, Olivier Gérard

nonn

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Vincenzo Librandi, Table of n, a(n) for n = 0..60
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A006116, A006117, A006118, A006119, A006120, A006121, A006122, A015195, A015196.

Row sums of triangle A022175.

Total/@Table[QBinomial[n, m, 11], {n, 0, 20}, {m, 0, n}] (* Vincenzo Librandi, Nov 02 2012 *)
Flatten[{1,RecurrenceTable[{a[n]==2*a[n-1]+(11^(n-1)-1)*a[n-2],a[0]==1,a[1]==2},a,{n,1,15}]}] (* Vaclav Kotesovec, Aug 21 2013 *)

a(n) = 2*a(n-1)+(11^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - Vaclav Kotesovec, Aug 21 2013

a(n) ~ c * 11^(n^2/4), where c = EllipticTheta[3,0,1/11]/QPochhammer[1/11,1/11] = 1.312069129398... if n is even and c = EllipticTheta[2,0,1/11]/QPochhammer[1/11,1/11] = 1.2291712170215... if n is odd. - Vaclav Kotesovec, Aug 21 2013

A135934 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - Fibonacci(k)*x).

Original entry on oeis.org

1, 1, 2, 4, 9, 24, 77, 299, 1419, 8312, 60452, 547939, 6213566, 88468601, 1585646789, 35846274127, 1023893974778, 37005881297226, 1694206791508891, 98335493373334998, 7241161595237290969, 676871453643079089963, 80351261743964014059133, 12117563014768206457325416

Offset: 0

Paul D. Hanna, Dec 07 2007

nonn

After the first term, row sums of triangle A111669. - Emanuele Munarini, Dec 05 2017

			A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-x)) + x^3/((1-x)*(1-x)*(1-2*x)) +
x^4/((1-x)*(1-x)*(1-2*x)(1-3*x)) + x^5/((1-x)*(1-x)*(1-2*x)*(1-3*x)*(1-5*x)) + x^6/((1-x)*(1-x)*(1-2*x)*(1-3*x)*(1-5*x)*(1-8*x)) +...

Alois P. Heinz, Table of n, a(n) for n = 0..140

Cf. A000045, A006116, A111669.

b:= proc(n, m) option remember; `if`(n=0, 1,
      (<<0|1>, <1|1>>^m)[1, 2]*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 08 2021

b[n_, m_] := b[n, m] = If[n == 0, 1,
   MatrixPower[{{0, 1}, {1, 1}}, m][[1, 2]]*b[n-1, m]+b[n-1, m+1]];
a[n_] :=  b[n, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 07 2022, after Alois P. Heinz *)

{a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-fibonacci(j)*x+x*O(x^n))), n)}

G.f.: (1 - G(0) )/(1-x) where G(k) = 1 - 1/(1-Fibonacci(k)*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 17 2013

G.f.: 1/(x*(1-x)*G(0)) - 1/x where G(k) = 1 - x/(x - 1/(1 + 1/(x*Fibonacci(k)-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 13 2013

A225609 Recurrence a(n) = 2^n*a(n-1) + a(n-2) with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 4, 33, 532, 17057, 1092180, 139816097, 35794013012, 18326674478241, 18766550459731796, 38433913668205196449, 157425329151518944386900, 1289628334843156860622681249, 21129270795495611155960953970516, 692363946716428521201685400328549537

Offset: 0

Vaclav Kotesovec, Aug 22 2013

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..81

Cf. A006116, A073587, A073588, A076131, A228465, A228467.

RecurrenceTable[{a[n]==2^n*a[n-1]+a[n-2],a[0]==0,a[1]==1},a,{n,0,15}]

a(n) ~ c * 2^(n^2/2 + n/2), where c = 0.52087674149344124670486211129137...

A228467 Recurrence a(n) = 2^n*a(n-1) - a(n-2) with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 4, 31, 492, 15713, 1005140, 128642207, 32931399852, 16860748082017, 17265373104585556, 35359467257443136671, 144832360621113983218860, 1186466662848698493085764449, 19439069659280715489603181513556, 636979433408843822314618558750438559

Offset: 0

Vaclav Kotesovec, Aug 22 2013

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..81

Cf. A225609, A076131, A073587, A073588, A228465, A006116.

RecurrenceTable[{a[n]==2^n*a[n-1]-a[n-2],a[0]==0,a[1]==1},a,{n,0,15}]
nxt[{n_,a_,b_}]:={n+1,b,b*2^(n+1)-a}; NestList[nxt,{1,0,1},20][[All,2]] (* Harvey P. Dale, Jul 02 2022 *)

a(n) ~ c * 2^(n^2/2 + n/2), where c = 0.4792100640621967581293535977698228585...

A381058 Irregular triangular array read by rows. Let S_n be the set of labeled graphs G on [n] with 2-colored nodes where black nodes are only connected to white nodes and vice versa. Orient the edges in each such graph G from black to white. T(n,k) is the number of graphs in S_n containing exactly k descents, n>=0, 0<=k<=A002620(n).

Original entry on oeis.org

1, 2, 5, 1, 16, 8, 2, 67, 56, 30, 8, 1, 374, 436, 358, 188, 68, 16, 2, 2825, 4143, 4508, 3460, 2032, 924, 320, 80, 13, 1, 29212, 50460, 66976, 66092, 52412, 34280, 18630, 8376, 3072, 892, 194, 28, 2, 417199, 811790, 1246486, 1471358, 1436404, 1195166, 859650, 537750, 292880, 138280, 56048, 19168, 5382, 1188, 192, 20, 1

Offset: 0

Geoffrey Critzer, Feb 12 2025

A descent in a labeled directed graph is an edge s->t such that s>t.

T(n,0) = A006116(n).

			    1;
    2;
    5,    1;
   16,    8,    2;
   67,   56,   30,    8,    1;
  374,  436,  358,  188,   68,  16,   2;
 2825, 4143, 4508, 3460, 2032, 924, 320, 80, 13, 1;
 ...

Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, arXiv:1909.01550 [math.CO], 20 Mar 2020.

Cf. A228890, A047863 (row sums), A006116, A111636, A381102, A381192.

nn = 7; B[n_] := FunctionExpand[QFactorial[n, (1 + u y)/(1 + y)]] (1+y)^Binomial[n,2]; e[z_] := Sum[z^n/B[n], {n, 0, nn}];Map[CoefficientList[#, u] &,  Table[B[n], {n, 0, nn}] CoefficientList[Series[e[z]^2, {z, 0, nn}],z] /. y -> 1] // Grid

A174526 Triangle t(n,m) = 2*A022166(n,m)-binomial(n,m), read by rows, 0<=m<=n.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 26, 64, 26, 1, 1, 57, 300, 300, 57, 1, 1, 120, 1287, 2770, 1287, 120, 1, 1, 247, 5313, 23587, 23587, 5313, 247, 1, 1, 502, 21562, 194254, 401504, 194254, 21562, 502, 1, 1, 1013, 86834, 1575986, 6619368, 6619368, 1575986, 86834

Offset: 0

Roger L. Bagula and Gary W. Adamson, Mar 21 2010

Row sums are 1, 2, 6, 24, 118, 716, 5586, 58296, 834142, 16566404, 459510186,... = 2*A006116(n)-2^n.

The column m=1 might be A000295. - R. J. Mathar, Nov 14 2011

			1;
1, 1;
1, 4, 1;
1, 11, 11, 1;
1, 26, 64, 26, 1;
1, 57, 300, 300, 57, 1;
1, 120, 1287, 2770, 1287, 120, 1;
1, 247, 5313, 23587, 23587, 5313, 247, 1;
1, 502, 21562, 194254, 401504, 194254, 21562, 502, 1;
1, 1013, 86834, 1575986, 6619368, 6619368, 1575986, 86834, 1013, 1;
1, 2036, 348457, 12695310, 107487764, 218443050, 107487764, 12695310, 348457, 2036, 1;

Cf. A008292.

A174526 := proc(n,k)
   2*A022166(n,k)-binomial(n,k) ;
end proc:
seq(seq(A174526(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Nov 14 2011

c[n_, q_] = Product[1 - q^i, {i, 1, n}];
t[n_, m_, q_] = 2*c[n, q]/(c[m, q]*c[n - m, q]) - Binomial[n, m];
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]
(* second program: *)
t[n_, m_] := 2 QBinomial[n, m, 2] - Binomial[n, m]; Table[t[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Apr 09 2016 *)

cp's OEIS Frontend

A015196 Sum of Gaussian binomial coefficients for q=10.

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A198260 Runs of 1s in binary strings corresponding to subgroups of nimber addition.

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A227960 Big equivalence classes (A227723) related to subgroups of nimber addition (A190939).

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A228465 Recurrence a(n) = a(n-1) + 2^n*a(n-2) with a(0)=0, a(1)=1.

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A015197 Sum of Gaussian binomial coefficients for q=11.

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A135934 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - Fibonacci(k)*x).

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A225609 Recurrence a(n) = 2^n*a(n-1) + a(n-2) with a(0)=0, a(1)=1.

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A228467 Recurrence a(n) = 2^n*a(n-1) - a(n-2) with a(0)=0, a(1)=1.

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