Bruce E. Sagan - cp's OEIS frontend

A140982 If (a_n) is a sequence then let L(a_n)=(b_n) where b_n = a_n^2 - a_{n-1} a_{n+1}. The given sequence is the rows of the triangle obtained by computing L^2(binomial(n,k)).

1, 3, 1, 6, 16, 1, 10, 100, 50, 1, 15, 400, 750, 120, 1, 21, 1225, 6125, 3675, 245, 1, 28, 3136, 34300, 54880, 13720, 448, 1, 36, 7056, 148176, 518616, 345744, 42336, 756, 1, 45, 14400, 529200, 3556224, 5186160, 1693440, 113400, 1200, 1, 55, 27225, 1633500

Offset: 3

Bruce E. Sagan, Jul 28 2008

L(binomial(n,k)) gives the Narayana numbers, A001263.

Peter R. W. McNamara and Bruce E. Sagan, Infinite log-concavity: developments and conjectures, arXiv:0808.1065 [math.CO], 2008-2009.
Peter R. W. McNamara and Bruce E. Sagan, Infinite log-concavity: developments and conjectures, Advances in Applied Mathematics, 44 (1) (2010), 1-15.

Cf. A001263.

a[n_, k_] := 2 * Binomial[n, k]^2 * Binomial[n, k - 1] * Binomial[n, k - 2] / ((n - 1) n^2); Table[ a[n, k], {n, 2, 11}, {k, 2, n}] // Flatten (* Robert G. Wilson v, Aug 03 2008 *)

a(n,k) = binomial(n,k)^2 * binomial(n,k-1) * binomial(n,k-2) / (n*binomial(n,2)).

More terms from Robert G. Wilson v, Aug 03 2008

A113248 Number of permutations pi in S_n such that maj pi and maj pi^(-1) have opposite parity where maj is the major index. Equivalently, the number of pi such that maj pi and inv pi have opposite parity where inv is the inversion number.

Original entry on oeis.org

0, 0, 2, 8, 56, 336, 2496, 19968, 181248, 1812480, 19956480, 239477760, 3113487360, 43588823040, 653836861440, 10461389783040, 177843708887040, 3201186759966720, 60822550111518720, 1216451002230374400

Offset: 0

Bruce E. Sagan, Oct 20 2005

nonn

a(2n) and a(2n+1) are both divisible by 2^n n! a(2n) = 2n a(2n-1) The number of pi in S_n such that maj pi is even and maj pi^(-1) is odd is exactly half of a(n)

			a(3)=2 because the following 2 permutations in S_3 have opposite parity for their major index and the major index of their inverse: 231, 312.

H. Barcelo, B. Sagan and S. Sundaram, Counting permutations by congruence class of major index, preprint, 2005.
H. Barcelo, B. Sagan and S. Sundaram, Counting permutations by congruence class of major index, Advances in Applied Mathematics, Volume 39, Issue 2, August 2007, Pages 269-281.

Cf. A113247.

a(2n) = 2 n^2 a(2n-2) + 2 n (n-1) b(2n-2) and a(2n+1) = 2 n (n+1) a(2n-1) + 2 n^2 b(2n-1) where b(n) is sequence A113247

A113247 Number of permutations pi in S_n such that maj pi and maj pi^(-1) have the same parity where maj is the major index. Equivalently, the number of pi such that maj pi and inv pi have the same parity where inv is the inversion number.

Original entry on oeis.org

1, 1, 2, 4, 16, 64, 384, 2544, 20352, 181632, 1816320, 19960320, 239523840, 3113533440, 43589468160, 653837506560, 10461400104960, 177843719208960, 3201186945761280, 60822550297313280, 1216451005946265600

Offset: 0

Bruce E. Sagan, Oct 20 2005

nonn

a(2n) and a(2n+1) are both divisible by 2^n n! a(2n) = 2n a(2n-1) The number of pi in S_n such that maj pi and maj pi^(-1) are both even is exactly half of a(n)

			a(3)=4 because the following 4 permutations in S_3 have the same parity for their major index and the major index of their inverse (and in fact are equal to their inverse): 123, 213, 321, 132.

H. Barcelo, B. Sagan and S. Sundaram, Counting permutations by congruence class of major index, preprint, 2005.
H. Barcelo, B. Sagan and S. Sundaram, Counting permutations by congruence class of major index, Advances in Applied Mathematics, Volume 39, Issue 2, August 2007, Pages 269-281.

Cf. A113248.

a(2n) = 2 n^2 a(2n-2) + 2 n (n-1) b(2n-2) and a(2n+1) = 2 n (n+1) a(2n-1) + 2 n^2 b(2n-1) where b(n) is sequence A113248

A089119 Complement of ((3A005836) union (3A005836 - 1) union (3*A005836 - 2)).

Original entry on oeis.org

4, 5, 6, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 31, 32, 33, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 85, 86, 87, 94, 95, 96, 97, 98, 99, 100, 101, 102

Offset: 1

Emeric Deutsch and Bruce E. Sagan, Dec 05 2003

nonn

Numbers k such that the Motzkin number A001006(k) == 0 (mod 3).

The asymptotic density of this sequence is 1 (Burns, 2016). - Amiram Eldar, Jan 30 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.

Cf. A001006, A005836.

(* m = MotzkinNumber *) m[0] = 1; m[n_] := m[n] = m[n - 1] + Sum[m[k]*m[n - 2 - k], {k, 0, n - 2}]; Select[Range[0, 120], Mod[m[#], 3] == 0 &] (* Jean-François Alcover, Jul 10 2013 *)

Offset corrected by Amiram Eldar, Jan 30 2021

A089118 Nonnegative numbers in (3*A005836 - 1) [A005836 are the numbers with base representation containing no 2].

Original entry on oeis.org

2, 8, 11, 26, 29, 35, 38, 80, 83, 89, 92, 107, 110, 116, 119, 242, 245, 251, 254, 269, 272, 278, 281, 323, 326, 332, 335, 350, 353, 359, 362, 728, 731, 737, 740, 755, 758, 764, 767, 809, 812, 818, 821, 836, 839, 845, 848, 971, 974, 980, 983, 998, 1001, 1007, 1010

Offset: 1

Emeric Deutsch and Bruce E. Sagan, Dec 05 2003

Numbers k such that the Motzkin number A001006(k) == 2 (mod 3).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.

Cf. A005836, A001006, A082575, A089119.

(* m = MotzkinNumber *) m[0] = 1; m[n_] := m[n] = m[n - 1] + Sum[ m[k]*m[n - 2 - k], {k, 0, n - 2}]; Select[ Range[1010], Mod[m[#], 3] == 2 &] (* Jean-François Alcover, Jul 10 2013 *)
Select[3*Range[350] - 1, DigitCount[# + 1, 3, 2] == 0 &] (* Amiram Eldar, Jun 04 2022 *)

Offset corrected to 1 by Jean-François Alcover, Jun 23 2016

A082575 Nonnegative numbers in (3A005836) union (3A005836 - 2) [A005836 lists the numbers with base-3 representation containing no 2].

Original entry on oeis.org

0, 1, 3, 7, 9, 10, 12, 25, 27, 28, 30, 34, 36, 37, 39, 79, 81, 82, 84, 88, 90, 91, 93, 106, 108, 109, 111, 115, 117, 118, 120, 241, 243, 244, 246, 250, 252, 253, 255, 268, 270, 271, 273, 277, 279, 280, 282, 322, 324, 325, 327, 331, 333, 334, 336, 349, 351, 352

Offset: 1

Emeric Deutsch and Bruce E. Sagan, Dec 05 2003

Numbers k such that the Motzkin number A001006(k) == 1 (mod 3).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.

Cf. A001006, A005836, A089118, A089119.

(* m = MotzkinNumber *) m[0] = 1; m[n_] := m[n] = m[n - 1] + Sum[m[k]*m[n - 2 - k], {k, 0, n - 2}]; Select[Range[0, 400], Mod[m[#], 3] == 1 &] (* Jean-François Alcover, Jul 10 2013 *)
max = 150; Sort @ Join[Select[3*Range[0, max], DigitCount[#, 3, 2] == 0 &], Select[3*Range[max] - 2, DigitCount[# + 2, 3, 2] == 0 &]] (* Amiram Eldar, Jun 04 2022 *)

Offset changed to 1 (sequence is a list) by L. Edson Jeffery, Nov 27 2015

A081706 Numbers n such that binary representation ends either in an odd number of ones followed by one zero or in an even number of ones.

Original entry on oeis.org

2, 3, 10, 11, 14, 15, 18, 19, 26, 27, 34, 35, 42, 43, 46, 47, 50, 51, 58, 59, 62, 63, 66, 67, 74, 75, 78, 79, 82, 83, 90, 91, 98, 99, 106, 107, 110, 111, 114, 115, 122, 123, 130, 131, 138, 139, 142, 143, 146, 147, 154, 155, 162, 163, 170, 171, 174, 175, 178, 179, 186

Offset: 1

Emeric Deutsch and Bruce E. Sagan, Apr 02 2003

Values of k such that the Motzkin number A001006(k) is even. Values of k such that the number of restricted hexagonal polyominoes with k+1 cells (A002212) is even.
Or union of sequences {2*A079523(n)+k}, k=0,1. A generalization see in comment to A161639. - Vladimir Shevelev, Jun 15 2009
Or intersection of sequences A121539 and {A121539(n)-1}. A generalization see in comment to A161890. - Vladimir Shevelev, Jul 03 2009
Also numbers n for which A010060(n+2) = A010060(n). - Vladimir Shevelev, Jul 06 2009
The asymptotic density of this sequence is 1/3 (Rowland and Yassawi, 2015; Burns, 2016). - Amiram Eldar, Jan 30 2021
Numbers of the form 4^k*(2*n-1)-2 and 4^k*(2*n-1)-1 where n and k are positive integers. - Michael Somos, Oct 22 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from G. C. Greubel)
Jean-Paul Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence, Journal de théorie des nombres de Bordeaux, Vol. 27, No. 2 (2015), pp. 375-388; arXiv preprint, arXiv:1401.3727 [math.NT], 2014.
Jean-Paul Allouche, André Arnold, Jean Berstel, Srećko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, A sequence related to that of Thue-Morse, Discrete Math., Vol. 139, No. 1-3 (1995), pp. 455-461.
Rob Burns, Asymptotic density of Motzkin numbers modulo small primes, arXiv:1611.04910 [math.NT], 2016.
Eric Rowland and Reem Yassawi, Automatic congruences for diagonals of rational functions, Journal de Théorie des Nombres de Bordeaux, Vol. 27, No. 1 (2015), pp. 245-288.
Index entries for 2-automatic sequences.

Cf. A001006, A002212, A003159, A010060, A079523, A121539, A161639, A161890.

(* m = MotzkinNumber *) m[0] = 1; m[n_] := m[n] = m[n - 1] + Sum[m[k]*m[n - 2 - k], {k, 0, n - 2}]; Select[Range[200], Mod[m[#], 2] == 0 &] (* Jean-François Alcover, Jul 10 2013 *)
Select[Range[200], EvenQ@Hypergeometric2F1[3/2, -#, 3, 4]&] (* Vladimir Reshetnikov, Nov 02 2015 *)

is(n)=valuation(bitor(n,1)+1,2)%2==0 \\ Charles R Greathouse IV, Mar 07 2013

from itertools import count, islice
def A081706_gen(): # generator of terms
    for n in count(0):
        if (n&-n).bit_length()&1:
            m = n<<2
            yield m-2
            yield m-1
A081706_list = list(islice(A081706_gen(),30)) # Chai Wah Wu, Jan 09 2023

def A081706(n):
    def f(x):
        c, s = (n+1>>1)+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1^1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    m, k = n+1>>1, f(n+1>>1)
    while m != k: m, k = k, f(k)
    return (m<<2)-1-(n&1) # Chai Wah Wu, Jan 29 2025

a(2n-1) = 2*A079523(n) = 4*A003159(n)-2; a(2n) = 4*A003159(n)-1.

Note that a(2n) = 1+a(2n-1).

A026845 Sum_{mu a partition of n} (f^mu/n!)^{-2} where f^mu is the number of standard Young tableaux of shape mu.

Original entry on oeis.org

1, 8, 81, 1424, 32152, 1144937, 53178768, 3360267976, 268737034880, 26735641360265, 3222856389284352, 463078022054303432, 78131995260953112576, 15295767841794798044432, 3438384401028669096232665, 879589866427669147125523584, 254053056142392070125392290952

Offset: 1

Bruce E. Sagan, Apr 06 2002

nonn

Arises from counting coverings of a genus g=2 Riemann surface - expansion of generating function A_g(q) = sum_{n>=0} a_{n,g} q^n where a_{n,g} = sum_{mu a partition of n} (f^mu/n!)^{2-2g}; note that A_0(q) = e^q and A_1(q) = prod_{i>=1} 1/(1-q^i).

Alois P. Heinz, Table of n, a(n) for n = 1..60

Cf. A047874. - Wouter Meeussen, Sep 30 2010

(* version 4.0 *) Needs["DiscreteMath`Combinatorica`"]; Table[Tr[(n!/ (NumberOfTableaux /@ Partitions[n]))^2],{n,20}] (* Wouter Meeussen, Sep 30 2010 *)

Terms 8 to 20 added by Wouter Meeussen, Sep 30 2010

cp's OEIS Frontend

User: Bruce E. Sagan

A140982 If (a_n) is a sequence then let L(a_n)=(b_n) where b_n = a_n^2 - a_{n-1} a_{n+1}. The given sequence is the rows of the triangle obtained by computing L^2(binomial(n,k)).

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A113248 Number of permutations pi in S_n such that maj pi and maj pi^(-1) have opposite parity where maj is the major index. Equivalently, the number of pi such that maj pi and inv pi have opposite parity where inv is the inversion number.

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A113247 Number of permutations pi in S_n such that maj pi and maj pi^(-1) have the same parity where maj is the major index. Equivalently, the number of pi such that maj pi and inv pi have the same parity where inv is the inversion number.

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A089119 Complement of ((3*A005836) union (3*A005836 - 1) union (3*A005836 - 2)).

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A089118 Nonnegative numbers in (3*A005836 - 1) [A005836 are the numbers with base representation containing no 2].

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A082575 Nonnegative numbers in (3*A005836) union (3*A005836 - 2) [A005836 lists the numbers with base-3 representation containing no 2].

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A081706 Numbers n such that binary representation ends either in an odd number of ones followed by one zero or in an even number of ones.

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A026845 Sum_{mu a partition of n} (f^mu/n!)^{-2} where f^mu is the number of standard Young tableaux of shape mu.

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A089119 Complement of ((3A005836) union (3A005836 - 1) union (3*A005836 - 2)).

A082575 Nonnegative numbers in (3A005836) union (3A005836 - 2) [A005836 lists the numbers with base-3 representation containing no 2].