A372368 -id:A372368 - cp's OEIS frontend

A100754 Triangle read by rows: T(n, k) = number of hill-free Dyck paths (i.e., no peaks at height 1) of semilength n and having k peaks.

1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 13, 29, 13, 1, 1, 19, 73, 73, 19, 1, 1, 26, 151, 266, 151, 26, 1, 1, 34, 276, 749, 749, 276, 34, 1, 1, 43, 463, 1781, 2762, 1781, 463, 43, 1, 1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1, 1, 64, 1093, 7253, 21659, 31004, 21659, 7253, 1093, 64, 1

Offset: 2

Emeric Deutsch, Jan 14 2005

Row n has n - 1 terms. Row sums yield the Fine numbers (A000957).
Related to the number of certain sets of non-crossing partitions for the root system A_n (p. 11, Athanasiadis and Savvidou). - Tom Copeland, Oct 19 2014
T(n,k) is the number of permutations pi of [n-1] with k - 1 descents such that s(pi) avoids the patterns 132, 231, and 312, where s is West's stack-sorting map. - Colin Defant, Sep 16 2018
The absolute values of the polynomials at -1 and j (cube root of 1) seem to be given by A126120 and A005043. - F. Chapoton, Nov 16 2021
Don Knuth observes that this sequence also arrises from the enumeration of restricted max-and-min-closed relations, only there it appears as an array read by antidiagonals: see the Knuth "Notes" link and A372068. Knuth also gives a formula expressing the array A372368 in terms of this array. He also reports that there is strong experimental evidence that the n-th term of row m in this array is a polynomial of degree 2*m-2 in n. - N. J. A. Sloane, May 12 2024

			T(4, 2) = 4 because we have UU*DDUU*DD, UU*DUU*DDD, UUU*DDU*DD and UUU*DU*DDD, where U = (1, 1), D = (1,-1) and * indicates the peaks.
Triangle starts:
   1;
   1,  1;
   1,  4,   1;
   1,  8,   8,    1;
   1, 13,  29,   13,    1;
   1, 19,  73,   73,   19,    1;
   1, 26, 151,  266,  151,   26,    1;
   1, 34, 276,  749,  749,  276,   34,   1;
   1, 43, 463, 1781, 2762, 1781,  463,  43,  1;
   1, 53, 729, 3758, 8321, 8321, 3758, 729, 53, 1;
   ...
As an array (for which the rows of the preceding triangle are the antidiagonals):
   1,  1,    1,     1,      1,      1,       1,        1,        1, ...
   1,  4,    8,    13,     19,     26,      34,       43,       53, ...
   1,  8,   29,    73,    151,    276,     463,      729,     1093, ...
   1, 13,   73,   266,    749,   1781,    3758,     7253,    13061, ...
   1, 19,  151,   749,   2762,   8321,   21659,    50471,   107833, ...
   1, 26,  276,  1781,   8321,  31004,   97754,   271125,   679355, ...
   1, 34,  463,  3758,  21659,  97754,  367285,  1196665,  3478915, ...
   1, 43,  729,  7253,  50471, 271125, 1196665,  4526470, 15118415, ...
   1, 53, 1093, 13061, 107833, 679355, 3478915, 15118415, 57500480, ...
   ...

Alois P. Heinz, Rows n = 2..142, flattened
C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024

Cf. A000957, A108263, A132081.

See also A099594, A272644, A372066, A372067, A372068.

T := (n, k) -> add((j/(n-j))*binomial(n-j, k-j)*binomial(n-j,k), j=0..min(k,n-k)): for n from 2 to 13 do seq(T(n, k), k = 1..n-1) od; # yields the sequence in triangular form

T[n_, k_] := Sum[(j/(n-j))*Binomial[n-j, k-j]*Binomial[n-j, k], {j, 0, Min[k, n-k]}]; Table[T[n, k], {n, 2, 13}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

T(n, k) = Sum_{j=0..min(k, n-k)} (j/(n-j)) * C(n-j, k-j) * C(n-j, k), n >= 2.
G.f.: t*z*r/(1 - t*z*r), where r = r(t, z) is the Narayana function defined by r = z*(1 + r)*(1 + t*r).
From Tom Copeland, Oct 19 2014: (Start)
With offset 0 for A108263 and offset 1 for A132081, row polynomials of this entry P(n, x) = Sum_{i} A108263(n, i)*x^i*(1 + x)^(n - 2*i) = Sum_{i} A132081(n - 2, i)*x^i*(1 + x)^(n - 2*i).
E.g., P(4, x) = 1*x*(1 + x)^(4 - 2*1) + 2*x^2*(1 + x)^(4 - 2*2) = x + 4*x^2 + x^3.
Equivalently, let Q(n, x) be the row polynomials of A108263. Then P(n, x) = (1 + x)^n * Q(n, x/(1 + x)^2).
E.g., P(4, x) = (1 + x)^4 * (x/(1 + x)^2 + 2 * (x/(1 + x)^2)^2).
See Athanasiadis and Savvidou (p. 7). (End)

A376839 a(1) = 1. For n > 1 if A007947(a(n-1)) is in A002110, a(n) is the smallest prime not already a term. Otherwise a(n) is the least novel multiple of the smallest non divisor prime of a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 10, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 23, 26, 27, 28, 30, 29, 32, 31, 34, 33, 36, 37, 38, 39, 40, 42, 25, 44, 45, 46, 48, 41, 50, 51, 52, 54, 43, 56, 57, 58, 60, 47, 62, 63, 64, 53, 66, 35, 68, 69, 70, 72, 59, 74, 75

Offset: 1

David James Sycamore, Oct 06 2024

nonn

A non divisor prime of a(n-1) is any prime p < Gpf(a(n-1)) which does not divide a(n-1). A007947(a(n-1)) is in A002110 iff a(n-1) is a term in A055932. Sequence is conjectured to be a permutation of the natural numbers (A000027) with primes in order.

Scatterplot shows trajectories of numbers whose smallest prime factor is prime p, e.g., for p = 5, numbers in A084967, p = 7, those in A084968, p = 11 those in A084969, etc. - Michael De Vlieger, Oct 09 2024

			a(1) = 1 = A002110(0) so a(2) = 2 (smallest prime not already a term).
a(2) = 2 = A002110(1) so a(3) = 3.
a(3) = 3 not a term in A002110 so a(4) is least novel multiple of 2, the least non divisor prime of 3. Therefore a(4) = 4 since 2 has occurred earlier.
a(39) = 42, not a term in A002110 so a(40) = 25, the least novel multiple of 5, the smallest non divisor prime of 42.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue or purple, with the latter additionally representing powerful numbers that are not prime powers.

Cf. A000040, A000027, A002110, A007947, A055932, A084967, A084968, A084969, A372368, A376865.

nn = 120; c[] := False; m[] := 1; f[x_] := FactorInteger[x][[All, 1]];
  Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; j = 2; v = 3;
  Do[If[Or[IntegerQ@ Log2[j], And[EvenQ[j], Union@ Differences@ PrimePi[#] == {1}]],
     k = v; While[c[k*m[k]], m[k]++]; k *= m[k],
     k = 2; While[Divisible[j, k], k = NextPrime[k]];
     While[c[k*m[k]], m[k]++]; k *= m[k]] &[f[j]];
  Set[{a[n], c[k], j}, {k, True, k}];
  If[k == v, While[c[v], v = NextPrime[v]]], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Oct 09 2024 *)

More terms from Michael De Vlieger, Oct 09 2024

A376865 a(1) = 1. Thereafter, if A007947(a(n-1)) is in A002110, a(n) = smallest prime not already a term (condition [A]). Otherwise, if q is the greatest prime < Gpf(a(n-1)) which does not divide a(n-1), a(n) is the least novel multiple of q,(condition [B]).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 14, 15, 12, 13, 22, 21, 20, 18, 17, 26, 33, 28, 25, 24, 19, 34, 39, 44, 35, 27, 16, 23, 38, 51, 52, 55, 42, 30, 29, 46, 57, 68, 65, 66, 49, 40, 36, 31, 58, 69, 76, 85, 78, 77, 45, 32, 37, 62, 87, 92, 95, 102, 91, 88, 56, 50, 48

Offset: 1

David James Sycamore, and Michael De Vlieger, Oct 07 2024

nonn

A sequence with 3 distinct phases, similar to A372368.
Define cycle c(i) to be a run of consecutive terms beginning with a prime a(n) = prime(i) resulting from condition [A], which ends when a(n) is a term in A055932.
Phase I consists of consecutive closed cycles c(i) that start with a(n) = prime(i) via condition [A] and end with a term in A055932. As n increases through cycle c(i), G = gpf(a(n)) strictly decreases, and g = gpf(m(q)) is small compared to G. This phase ends at n = 4318.
Phase II consists of closed cycles c(i) that start with a(n) = prime(j), j > i, via condition [A] and end with a term in A055932. As n increases through cycle c(i), at times, g > G and we have a rejuvenated cycle. We may see multiple condition [B] primes, as well as runs of composite a(n) for n = 99528..155219 and n = 222811..262605. The humps in scatterplot are associated with these particular runs of composite terms. Rejuvenation of a cycle has G increment m(q) each time. A "ridge" of high m(q) values builds and grows increasingly difficult to traverse to reach G = 11, where we might have a number in A055932 and close the cycle. This phase likely ends with n = 2048704.
Phase III consists of condition [A] prime a(2048704) = prime(742) = 5647 and terms that follow, starting cycle c(135). As n increases, there are repeated rejuvenations and regular entry of primes through condition [B]. The repeated rejuvenations increase and expand a bank of high values of m(q) across many primes q only a few dozen iterations after new primes appear. New primes mean that prime(i) increases while in order to find a(n) in A055932, we need numbers with G = 11. Therefore, the circumstance that needs to arise to close the cycle becomes harder to achieve as n increases.
It is unlikely that this sequence is a permutation of natural numbers.
A full description of the phased behavior of this sequence is given in the link.

			a(1) = 1 = A002110(0), so a(2) = 2 (smallest prime not already a term).
a(2) = 2 = A002110(1), so a(3) = 3.
a(3) = 3 not in A002110 therefore a(4) = 4, the least novel multiple of 2.
rad(4) = 2 so a(5) = 5 the smallest prime not already a term.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of G(n) = gpf(a(n)), n = 1..3000000, showing the confinement of G(n) to large primes for n > 2048704.
Michael De Vlieger, Notes on this sequence.
Michael De Vlieger, Tipping Point Sequence A365865, ResearchGate, 2024.

Cf. A000040, A000027, A002110, A007947, A055932, A376839, A372368.

nn = 120; c[] := False; m[] := 1; f[x_] := FactorInteger[x][[All, 1]];
Array[Set[{a[#], c[#], m[#]}, {#, True, 2}] &, 2]; j = 2; v = 3;
Do[If[Or[IntegerQ@ Log2[j], And[EvenQ[j], Union@ Differences@ PrimePi[#] == {1}]],
     k = v; While[c[k*m[k]], m[k]++]; k *= m[k],
     k = Last[#]; While[And[k > 1, Divisible[j, k]], k = NextPrime[k, -1]];
     While[c[k*m[k]], m[k]++]; k *= m[k]] &[f[j]];
  Set[{a[n], c[k], j}, {k, True, k}];
  If[k == v, While[c[v], v = NextPrime[v] ] ], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Oct 09 2024 *)

More terms from Michael De Vlieger, Oct 09 2024

cp's OEIS Frontend

A100754 Triangle read by rows: T(n, k) = number of hill-free Dyck paths (i.e., no peaks at height 1) of semilength n and having k peaks.

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A376839 a(1) = 1. For n > 1 if A007947(a(n-1)) is in A002110, a(n) is the smallest prime not already a term. Otherwise a(n) is the least novel multiple of the smallest non divisor prime of a(n-1).

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A376865 a(1) = 1. Thereafter, if A007947(a(n-1)) is in A002110, a(n) = smallest prime not already a term (condition [A]). Otherwise, if q is the greatest prime < Gpf(a(n-1)) which does not divide a(n-1), a(n) is the least novel multiple of q,(condition [B]).

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