A131924 -id:A131924 - cp's OEIS frontend

A131923 Triangle read by rows: T(n,k) = binomial(n,k) + n.

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 10, 8, 5, 6, 10, 15, 15, 10, 6, 7, 12, 21, 26, 21, 12, 7, 8, 14, 28, 42, 42, 28, 14, 8, 9, 16, 36, 64, 78, 64, 36, 16, 9, 10, 18, 45, 93, 135, 135, 93, 45, 18, 10, 11, 20, 55, 130, 220, 262, 220, 130, 55, 20, 11, 12, 22, 66

Offset: 0

Gary W. Adamson, Jul 29 2007

Row sums = A131924: (1, 4, 10, 20, 36, 62, 106, 184, ...).

			First few rows of the triangle are:
   1;
   2,   2;
   3,   4,   3;
   4,   6,   6,   4;
   5,   8,  10,   8,   5;
   6,  10,  15,  15,  10,   6;
   7,  12,  21,  26,  21,  12,   7;
   8,  14,  28,  42,  42,  28,  14,   8;
   9,  16,  36,  64,  78,  64,  36,  16,   9;
  10,  18,  45,  93, 135, 135,  93,  45,  18,  10;
  ...

Muniru A Asiru, Table of n, a(n) for n = 0..5049

Cf. A002024, A007318, A000012, A003056, A131924 (row sums), A003991.

a:=Flat(List([0..10],n->List([0..n],k->Binomial(n,k)+n))); # Muniru A Asiru, Jul 16 2018

/* As triangle */ [[Binomial(n, k) + n: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 17 2018

T[n_, m_] = Binomial[n, m] + n; Table[Table[T[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%] (* Roger L. Bagula, Jul 30 2008 *)

T(n,k) = binomial(n,k) + n \\ Charles R Greathouse IV, Oct 16 2013

A007318 + A002024 - A000012 = A007318 + A003056 as infinite lower triangular matrices. A002024 = (1; 2,2; 3,3,3;...); A007318 = Pascal's triangle and A000012 = (1; 1,1; 1,1,1;...).

Edited, changing formula by Roger L. Bagula, Jul 30 2008
New name from Franklin T. Adams-Watters, Oct 16 2013
Terms 54 onwards from Muniru A Asiru, Jul 16 2018

A295077 a(n) = 2n(n-1) + 2^n - 1.

Original entry on oeis.org

0, 1, 7, 19, 39, 71, 123, 211, 367, 655, 1203, 2267, 4359, 8503, 16747, 33187, 66015, 131615, 262755, 524971, 1049335, 2097991, 4195227, 8389619, 16778319, 33555631, 67110163, 134219131, 268436967, 536872535, 1073743563, 2147485507

Offset: 0

Franck Maminirina Ramaharo, Nov 13 2017

We have a(0) = 0, and for n > 0, a(n) is a subsequence of A131098 where the indices are given by the partial sums of A288382.
For n > 0, a(n) gives the number of words of length n over the alphabet A = {a,b,c,d} such that: a word containing 'c' does not contain 'b' or 'd'; a word cannot be fully written with 'a'; a word contains letters from {b,d} if and only if it contains exactly a unique couple of letters from {b,d}. Thus a(1) = 1 where the corresponding word is "c" since 'c' is the only letter allowed to be written alone.
Primes in the sequence are 7, 19, 71, 211, 367, 2267, 16747, 524971, ... which are of the form 4*k + 3 (A002145).
The second difference of this sequence is A140504.

			a(4) = 39. The corresponding words are aabb, aabd, aadb, aadd, abab, abad, abba, abda, adab, adad, adba, adda, aaac, aaca, aacc, acaa, acac, acca, accc, baab, baad, baba, bada, bbaa, bdaa, caaa, caac, caca, cacc, ccaa, ccac, ccca, cccc, daab, daad, daba, dada, dbaa, ddaa.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..70 from Franck Maminirina Ramaharo)
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2)

Cf. A000079, A002145, A008586, A046092, A126646, A131098, A131924, A140504, A288382.

[2*n*(n-1)+2^n-1 : n in [0..40]]; // Wesley Ivan Hurt, Nov 26 2017

A295077:=n->2*n*(n-1)+2^n-1; seq(A295077(n), n=0..70);

Table[2 n (n - 1) + 2^n - 1, {n, 0, 70}]

a(n) = 2*n*(n-1) + 2^n - 1; \\ Michel Marcus, Nov 14 2017

G.f.: (x + 2*x^2 - 7*x^3)/((1 - x)^3*(1 - 2*x)).
a(0)=0, a(1)=1, a(2)=7, a(3)=19; for n>3, a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4).
a(n) = 2*A131924(n-1) - 1 for n>0, a(0)=0.
a(n) = a(n-1) + A000079(n-1) + A008586(n-1) for n>0, a(0)=0.
a(n) = A126646(n-1) + A046092(n-1) for n>0, a(0)=0.
a(n+1) - 2*a(n) + a(n-1) = A140504(n-1) for n>0, a(0)=0.
E.g.f.: exp(2*x) - (1 - 2*x^2)*exp(x). - G. C. Greubel, Oct 17 2018

A366542 Number of discrete uninorms defined on the finite chain L_n={0,1,...n}, U:L_n^2->L_n, whose underlying operators are smooth and idempotent, or smooth and idempotent-free.

Original entry on oeis.org

2, 6, 14, 30, 56, 100, 178, 322, 596, 1128, 2174, 4246, 8368, 16588, 33002, 65802, 131372, 262480, 524662, 1048990, 2097608, 4194804, 8389154, 16777810, 33555076, 67109560, 134218478, 268436262, 536871776, 1073742748, 2147484634, 4294968346, 8589935708, 17179870368, 34359739622

Offset: 1

Marc Munar, Oct 12 2023

The number of discrete uninorms defined on the finite chain L_n={0,1,...n} whose underlying operators are smooth and idempotent or smooth and idempotent-free, i.e., the number of monotonic increasing binary functions U:L_n^2->L_n such that U is associative (U(x,U(y,z))=U(U(x,y),z) for all x,y,z in L_N), U is commutative (U(x,y)=U(y,x) for all x,y in L_n) and has some neutral element e in L_n (U(x,e)=U(e,x)=x for all x in L_n), such that U restricted to {0,...,e} and to {e,...,n} is smooth and idempotent, or smooth and idempotent-free.

D. Ruiz-Aguilera and J. Torrens, A characterization of discrete uninorms having smooth underlying operators, Fuzzy Sets and Systems, Volume 268, 2015, 44-58.
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).

Cf. A131924.

Join[{1, 6}, Table[2^n + n + n^2 - 6, {n, 3, 35}]]

a(1)=2, a(2)=6 and a(n) = 2^n+n*(n+1) - 6 for n>=3.
From Stefano Spezia, Nov 05 2023: (Start)
G.f.: 2*x*(1 - 2*x + x^2 - 3x^4 + 2*x^5)/((1 - x)^3*(1 - 2*x)).
a(n) = A131924(n) - 6 for n>=3. (End)

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A131923 Triangle read by rows: T(n,k) = binomial(n,k) + n.

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A295077 a(n) = 2*n*(n-1) + 2^n - 1.

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A366542 Number of discrete uninorms defined on the finite chain L_n={0,1,...n}, U:L_n^2->L_n, whose underlying operators are smooth and idempotent, or smooth and idempotent-free.

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Comments

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Mathematica

Formula

A295077 a(n) = 2n(n-1) + 2^n - 1.