A100071 -id:A100071 - cp's OEIS frontend

A191793 Triangle read by rows: T(n,k) is the number of length n left factors of Dyck paths having k UUDD's, where U=(1,1) and D=(1,-1).

1, 1, 2, 3, 5, 1, 8, 2, 14, 6, 23, 12, 41, 28, 1, 69, 54, 3, 124, 116, 12, 212, 220, 30, 383, 453, 87, 1, 662, 852, 198, 4, 1200, 1712, 500, 20, 2091, 3204, 1080, 60, 3799, 6338, 2526, 206, 1, 6661, 11824, 5280, 540, 5, 12122, 23136, 11772, 1560, 30, 21359, 43068, 24066, 3780, 105

Offset: 0

Emeric Deutsch, Jun 18 2011

Row n contains 1+floor(n/4) entries.
Sum of entries in row n is binomial(n, floor(n/2)) =A001405(n).
T(n,0)=A191794(n).
Sum(k*T(n,k), k>=0)=A100071(n-3).

			T(9,2)=3 because we have (UUDD)(UUDD)U, (UUDD)U(UUDD), and U(UUDD)(UUDD), where U=(1,1) and D=(1,-1) (the UUDD's are shown between parentheses).
Triangle starts:
1;
1;
2;
3;
5,1;
8,2;
14,6;
23,12;
41,28,1;

Cf. A001405, A191794

eq := z^2*C^2-(1+z^4-t*z^4)*C+1 = 0: C := RootOf(eq, C): G := C/(1-z*C): Gser := simplify(series(G, z = 0, 23)): for n from 0 to 19 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 19 do seq(coeff(P[n], t, k), k = 0 .. floor((1/4)*n)) end do; # yields sequence in triangular form

G.f.: G(t,z)= C/(1-z*C), where C=C(t,z) is given by z^2*C^2-(1+z^4-t*z^4)*C +1=0.

A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

Original entry on oeis.org

1, 1, 3, 17, 155, 2677, 73327, 3578339, 329652351

Offset: 1

Thomas Scheuerle, Aug 19 2022

This sequence can be calculated by a recursive algorithm:
Let B1 be an array of finite length, the "1" denotes that it is the first generation. Let B1' be the reversed version of B1. Let C be the element-wise product C = B1 * B1'. Then B2 is a concatenation of taking each element of B1 and add all divisors of the corresponding element in C. If we start with B1 = {1} then we get this sequence of arrays: B2 = {2}, B3 = {3, 4, 6}, ... . a(n) is the length of the array Bn. In short the length of Bn+1 and so a(n+1) is the sum over A000005(Bn * Bn').
The transform used in the definition of this sequence is its own inverse, so if c = S(b) then b = S(c). The eigensequence is 2^n = S(2^n).
There exist some transformation pairs of infinite sequences in the database:
  A000027 <--> A000142; A000079 <--> A000079; A000045 <--> A001654;
  A026549 <--> A038754; A100071 <--> A001405; A058295 <--> A------;
  A111286 <--> A098011; A093968 <--> A205825; A166447 <--> A------;
  A098558 <--> A004277; A010551 <--> A087811; A100538 <--> A329227;
  A079352 <--> A------; A082458 <--> A------; A008233 <--> A264635;
  A138278 <--> A------; A006501 <--> A264557; A336496 <--> A------;
  A019464 <--> A------; A062112 <--> A------; A171647 <--> A359039;
  A279312 <--> A------; A031923 <--> A------.
These transformation pairs are conjectured:
  A137326 <--> A------; A066332 <--> A300902; A208147 <--> A308546;
  A057895 <--> A------; A349080 <--> A------; A019442 <--> A------;
  A349079 <--> A------.
("A------" means not yet in the database.)
Some sequences in the lists above may need offset adjustment to force a beginning with 1,2,... in the transformation.
If we allowed signed rational numbers, further interesting transformation pairs could be observed. For example, 1/n will transform into factorials with alternating sign. 2^(-n) transforms into ones with alternating sign and 1/A000045(n) into A000045 with alternating sign.

			a(4) = 17. The 17 transformation pairs of length 4 are:
  {1, 2, 3, 4}  = S({1, 2, 6, 24}).
  {1, 2, 3, 5}  = S({1, 2, 6, 15}).
  {1, 2, 3, 6}  = S({1, 2, 6, 12}).
  {1, 2, 3, 9}  = S({1, 2, 6, 9}).
  {1, 2, 3, 12} = S({1, 2, 6, 8}).
  {1, 2, 3, 21} = S({1, 2, 6, 7}).
  {1, 2, 4, 5}  = S({1, 2, 4, 20}).
  {1, 2, 4, 6}  = S({1, 2, 4, 12}).
  {1, 2, 4, 8}  = S({1, 2, 4, 8}).
  {1, 2, 4, 12} = S({1, 2, 4, 6}).
  {1, 2, 4, 20} = S({1, 2, 4, 5}).
  {1, 2, 6, 7}  = S({1, 2, 3, 21}).
  {1, 2, 6, 8}  = S({1, 2, 3, 12}).
  {1, 2, 6, 9}  = S({1, 2, 3, 9}).
  {1, 2, 6, 12} = S({1, 2, 3, 6}).
  {1, 2, 6, 15} = S({1, 2, 3, 5}).
  {1, 2, 6, 24} = S({1, 2, 3, 4}).
b(1) = 1 by definition, b(2) = 1+1 as 1 has only 1 as divisor.
a(3) = A000005(b(2)*b(2)) = 3.
The divisors of b(2) are 1,2,4. So b(3) can be b(2)+1, b(2)+2 and b(2)+4.
a(4) = A000005((b(2)+1)*(b(2)+4)) + A000005((b(2)+2)*(b(2)+2)) + A000005((b(2)+4)*(b(2)+1)) = 17.

Cf. A000005.
Cf. A000027, A000045, A000079, A000142, A001405.
Cf. A001654, A004277, A006501, A008233, A019442.
Cf. A010551, A019464, A026549, A031923, A038754.
Cf. A057895, A058295, A062112, A066332, A100071.
Cf. A079352, A082458, A087811, A093968, A098011.
Cf. A098558, A100538, A111286, A166447, A137326.
Cf. A138278, A171647, A205825, A208147, A264557.
Cf. A264635, A308546, A329227, A336496, A349079.
Cf. A349080, A359039.

A242172 a(n) = 2^n*binomial((n + 1 + (n mod 2))/2, 1/2).

Original entry on oeis.org

1, 3, 6, 15, 30, 70, 140, 315, 630, 1386, 2772, 6006, 12012, 25740, 51480, 109395, 218790, 461890, 923780, 1939938, 3879876, 8112468, 16224936, 33801950, 67603900, 140408100, 280816200, 581690700, 1163381400, 2404321560, 4808643120, 9917826435, 19835652870

Offset: 0

Peter Luschny, May 06 2014

nonn

Cf. A100071, A002457, A033876.

a := n -> 2^n*binomial((n+1+(n mod 2))/2, 1/2); seq(a(n), n=0..29);

a[n_] := 2^n*Binomial[(n + 1 + Mod[n, 2])/2, 1/2]; Array[a, 33, 0] (* Amiram Eldar, Mar 04 2023 *)

a(2*n) = A002457(n).
a(2*n+1) = A033876(n).
a(2*n+2)/2 = a(2*n+1).
Conjecture: (n+1)*a(n) -2*a(n-1) +4*(-n-1)*a(n-2)=0. - R. J. Mathar, May 11 2014
a(n) = A100071(n+2)/2. - Michel Marcus, Sep 14 2015
Sum_{n>=0} 1/a(n) = 2*Pi/sqrt(3) - 2. - Amiram Eldar, Mar 04 2023
a(n) = (n+2)*binomial(n+1,ceiling(n/2))/2. - Wesley Ivan Hurt, Nov 23 2023

A348864 a(n) is the number of multiplications required to compute the permanent of general n X n matrices using trellis method with normalization.

Original entry on oeis.org

0, 4, 12, 32, 70, 162, 350, 800, 1746, 3950, 8602, 19164, 41392, 90846, 194490, 421568, 895594, 1922022, 4057298, 8638580, 18140640, 38378054, 80244562, 168877272, 351827100, 737208082, 1531123830, 3196464740, 6621247636, 13779365430, 28477354354, 59102191488, 121898268954

Offset: 1

Stefano Spezia, Nov 02 2021

nonn

Han Mao Kiah, Alexander Vardy and Hanwen Yao, Computing Permanents on a Trellis, arXiv:2107.07377 [cs.IT], 2021.

Cf. A001787, A002378, A058877, A100071.

a[n_]:=n 2^(n-1)-Ceiling[n/2]Binomial[n,Floor[n/2]]+n^2-n; Array[a,33]

a(n) = n*2^(n-1) - ceil(n/2)*binomial(n, floor(n/2)) + n^2 - n; \\ Michel Marcus, Nov 03 2021

a(n) = n*2^(n-1) - ceiling(n/2)*binomial(n, floor(n/2)) + n^2 - n (see Theorem 6, p. 11 in Kiah et al.).
a(n) = A001787(n) - A100071(n) + A002378(n-1).
O.g.f.: x*(1/(1 - 2*x)^2 + 2*x/(1 - x)^3 - 1/((1 - 2*x)*sqrt(1 - 4*x^2))).
E.g.f.: exp(x)*x*(exp(x) + x) - (1 + x)*BesselI(1, 2*x) - x*BesselI(2, 2*x).
D-finite with recurrence (n-1)*(n-2)*(n-4)*(3*n-23)*a(n) -3*(n -2)*(3*n^3-34*n^2+91*n-20)*a(n-1) -2*(n-1)*(n-3)*(3*n^2 -47*n+164)*a(n-2) +12*(3*n-22)*(n-1)*(n-2)*(n-4)*a(n-3) -8*(3*n-20)*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Mar 06 2022

cp's OEIS Frontend

A191793 Triangle read by rows: T(n,k) is the number of length n left factors of Dyck paths having k UUDD's, where U=(1,1) and D=(1,-1).

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A356639 Number of integer sequences b with b(1) = 1, b(m) > 0 and b(m+1) - b(m) > 0, of length n which transform under the map S into a nonnegative integer sequence. The transform c = S(b) is defined by c(m) = Product_{k=1..m} b(k) / Product_{k=2..m} (b(k) - b(k-1)).

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A242172 a(n) = 2^n*binomial((n + 1 + (n mod 2))/2, 1/2).

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A348864 a(n) is the number of multiplications required to compute the permanent of general n X n matrices using trellis method with normalization.

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