A037955 -id:A037955 - cp's OEIS frontend

A000594 Ramanujan's tau function (or Ramanujan numbers, or tau numbers).

1, -24, 252, -1472, 4830, -6048, -16744, 84480, -113643, -115920, 534612, -370944, -577738, 401856, 1217160, 987136, -6905934, 2727432, 10661420, -7109760, -4219488, -12830688, 18643272, 21288960, -25499225, 13865712, -73279080, 24647168

Offset: 1

N. J. A. Sloane

Coefficients of the cusp form of weight 12 for the full modular group.
It is conjectured that tau(n) is never zero (this has been verified for n < 816212624008487344127999, see the Derickx, van Hoeij, Zeng reference).
M. J. Hopkins mentions that the only known primes p for which tau(p) == 1 (mod p) are 11, 23 and 691, that it is an open problem to decide if there are infinitely many such p and that no others are known below 35000. Simon Plouffe has now searched up to tau(314747) and found no other examples. - N. J. A. Sloane, Mar 25 2007
Number 1 of the 74 eta-quotients listed in Table I of Martin (1996).
With Dedekind's eta function and the discriminant Delta one has eta(z)^24 = Delta(z)/(2*Pi)^12 = Sum_{m >= 1} tau(m)*q^m, with q = exp(2*Pi*i*z), and z in the complex upper half plane, where i is the imaginary unit. Delta is the eigenfunction of the Hecke operator T_n (n >= 1) with eigenvalue tau(n): T_n Delta = tau(n) Delta. From this the formula for tau(m)*tau(n) given below in the formula section follows. See, e.g., the Koecher-Krieg reference, Lemma and Satz, p. 212. Or the Apostol reference, eq. (3) on p. 114 and the first part of section 6.13 on p. 131. - Wolfdieter Lang, Jan 26 2016
For the functional equation satisfied by the Dirichlet series F(s), Re(s) > 7, of a(n) see the Hardy reference, p. 173, (10.9.4). It is (2*Pi)^(-s) * Gamma(s) * F(s) = (2*Pi)^(s-12) * Gamma(12-s) * F(12-s). This is attributed to J. R. Wilton, 1929, on p. 185. - Wolfdieter Lang, Feb 08 2017
Conjecture: |a(n)| with n > 1 can never be a perfect power. This has been verified for n up to 10^6. - Zhi-Wei Sun, Dec 18 2024
Conjecture: The numbers |a(n)| (n = 1,2,3,...) are distinct. This has been verified for the first 10^6 terms. - Zhi-Wei Sun, Dec 21 2024
Conjecture: |a(n)| > 2*n^4 for all n > 2. This has been verified for n = 3..10^6. - Zhi-Wei Sun, Dec 25 2024
Conjecture: a(m)^2 + a(n)^2 can never be a perfect power. This implies Lehmer's conjecture that a(n) is never zero. We have verified that there is no perfect power among a(m)^2 + a(n)^2 with m,n <= 1000 . - Zhi-Wei Sun, Dec 28 2024
Conjecture: The equation |a(m)a(n)| = x^k with m < n, k > 1 and x >= 0 has no solution. This has been verified for m < n <= 5000. - Zhi-Wei Sun, Dec 29 2024
For some conjectures motivated by additive combinatorics, one may consult the link to Question 485138 at MathOverflow. - Zhi-Wei Sun, Jan 25 2025

			G.f. = q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 + ...
35328 = (-24)*(-1472) = a(2)*a(4) = a(2*4) + 2^11*a(2*4/4) = 84480 + 2048*(-24) = 35328. See a comment on T_n Delta = tau(n) Delta above. - _Wolfdieter Lang_, Jan 21 2016

Tom M. Apostol, Modular functions and Dirichlet series in number theory, second Edition, Springer, 1990, pp. 114, 131.
Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, AMS 2001; see p. 298.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (32.2).
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, lecture X, pp. 161-185.
Bruce Jordan and Blair Kelly (blair.kelly(AT)att.net), The vanishing of the Ramanujan tau function, preprint, 2001.
Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 210 - 212.
Yu. I. Manin, Mathematics and Physics, Birkhäuser, Boston, 1981.
Henry McKean and Victor Moll, Elliptic Curves, Camb. Univ. Press, 1999, p. 139.
M. Ram Murty, The Ramanujan tau-function, pp. 269-288 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
Srinivasa Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 153, Ed. G. H. Hardy et al., AMS Chelsea 2000.
Srinivasa Ramanujan, On Certain Arithmetical Functions. Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
Jean-Pierre Serre, A course in Arithmetic, Springer-Verlag, 1973, see p. 98.
Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, 1994, see p. 482.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. P. F. Swinnerton-Dyer, Congruence properties of tau(n), pp. 289-311 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
Don Zagier, Introduction to Modular Forms, Chapter 4 in M. Waldschmidt et al., editors, From Number Theory to Physics, Springer-Verlag, 1992.
Don Zagier, "Elliptic modular forms and their applications", in: The 1-2-3 of modular forms, Springer Berlin Heidelberg, 2008, pp. 1-103.

Simon Plouffe, Table of n, a(n) for n = 1..16090
Jennifer S. Balakrishnan, William Craig, and Ken Ono, Variations of Lehmer's Conjecture for Ramanujan's tau-function, arXiv:2005.10345 [math.NT], 2020.
Jennifer S. Balakrishnan, Ken Ono, and Wei-Lun Tsai, Even values of Ramanujan's tau-function, arXiv:2102.00111 [math.NT], 2021.
Bruce C. Berndt and Ken Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary.
Bruce C. Berndt and Ken Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary, Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp.
Bruce C. Berndt and Pieter Moree, Sums of two squares and the tau-function: Ramanujan's trail, arXiv:2409.03428 [math.NT], 2024.
Matthew Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory, Vol. 98, No. 2 (2003), 377-389. MR1955423 (2003k:11071).
François Brunault, La fonction Tau de Ramanujan. [Wayback Machine link]
Denis Xavier Charles, Computing The Ramanujan Tau Function.
Benoit Cloitre, On the fractal behavior of primes, 2011.
John Cremona, Home page.
Maarten Derickx, Mark van Hoeij, and Jinxiang Zeng, Computing Galois representations and equations for modular curves X_H(l), arXiv:1312.6819 [math.NT], 2013-2014.
Bas Edixhoven, Jean-Marc Couveignes, Robin de Jong, Franz Merkl, and Johan Bosman, Computing the coefficients of a modular form, arXiv:math/0605244 [math.NT], 2006-2010.
John A. Ewell, Ramanujan's Tau Function, Proc. Amer. Math. Soc. 128 (2000), 723-726.
John A. Ewell, Ramanujan's Tau Function.
Steven R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]
Luis H. Gallardo, On some formulae for Ramanujan's tau function, Rev. Colomb. Matem. 44 (2010) 103-112.
M. Z. Garaev, V. C. Garcia, and S. V. Konyagin, Waring problem with the Ramanujan tau function, Izvestiya: Mathematics, Vol. 72, No 1 (2008), pp. 35-46; arXiv preprint, arXiv:math/0607169 [math.NT], 2006.
Frank Garvan and Michael J. Schlosser, Combinatorial interpretations of Ramanujan's tau function, Discrete Mathematics, Vol. 341, No. 10 (2018), pp. 2831-2840; arXiv preprint, arXiv:1606.08037 [math.CO], 2016.
Hansraj Gupta, The Vanishing of Ramanujan's Function(n), Current Science, 17 (1948), p. 180. [Wayback Machine link]
James Lee Hafner and Jeffrey Stopple, A Heat Kernel Associated to Ramanujan's Tau Function, The Ramanujan Journal, Vol. 4, No. 2 (2000), pp. 123-128.
Yang-Hui He and John McKay, Moonshine and the Meaning of Life, in: M. Bhargava et al., Finite Simple Groups: Thirty Years of the Atlas and Beyond, Contemporary Mathematics, Vol. 694, American Mathematical Society, 2017; arXiv preprint, arXiv:1408.2083 [math.NT], 2014.
Michael J. Hopkins, Algebraic topology and modular forms, Proc. Internat. Congress Math., Beijing 2002, Vol. I, pp. 291-317; arXiv preprint, arXiv:math/0212397 [math.AT], 2002.
Masanobu Kaneko and Don Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998
Jon Keating and Brady Haran, The Key to the Riemann Hypothesis, Numberphile video (2016).
Jerry B. Keiper, Ramanujan's Tau-Dirichlet Series [Dead link?]
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
N. Laptyeva and V. K. Murty, Fourier coefficients of forms of CM-type, Indian Journal of Pure and Applied Mathematics, Volume 45, Issue 5 (October 2014), pp. 747-758.
D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433.
D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy]
D. H. Lehmer, Tables of Ramanujan's function tau(n), Math. Comp., 24 (1970), 495-496.
LMFDB, Newform orbit 1.12.a.a
Florian Luca and Igor E. Shparlinski, Arithmetic properties of the Ramanujan function, Proceedings of the Indian Academy of Sciences-Mathematical Sciences, Vol. 116. No. 1 (2006), pp. 1-8; arXiv preprint, arXiv:math/0607591 [math.NT], 2006.
Nik Lygeros and Olivier Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq. 13 (2010), Article 10.7.4.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc., Vol. 348, No. 12 (1996), 4825-4856, see page 4852 Table I.
Yuri Matiyasevich, Computational rediscovery of Ramanujan's tau numbers, Integers (2018) 18A, Article #A14.
Keith Matthews, Computing Ramanujan's tau function.
Stephen C. Milne, New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan's tau function, Proc. Nat. Acad. Sci. USA, 93 (1996) 15004-15008.
Stephen C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
Louis J. Mordell, On Mr. Ramanujan's empirical expansions of modular functions, Proceedings of the Cambridge Philosophical Society 19 (1917), pp. 117-124.
Pieter Moree, On some claims in Ramanujan's 'unpublished' manuscript on the partition and tau functions, arXiv:math/0201265 [math.NT], 2002.
M. Ram Murty and V. Kumar Murty, The Ramanujan tau-function, in: The mathematical legacy of Srinivasa Ramanujan (Springer, 2012), p 11-23.
M. Ram Murty, V. Kumar Murty, and T. N. Shorey, Odd values of the Ramanujan tau-function, Bulletin de la S. M. F., tome 115 (1987), p. 391-395.
Douglas Niebur, A formula for Ramanujan's tau-function, Illinois Journal of Mathematics, vol.19, no.3, pp.448-449, (1975). - _Joerg Arndt_, Sep 06 2015
Oklahoma State Mathematics Department, Ramanujan tau L-Function. [broken link]
Jon Perry, Ramanujan's Tau Function. [Wayback Machine link]
Simon Plouffe, The first 225035 terms (432 MB) [Wayback Machine link]
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.
Srinivasa Ramanujan, Collected Papers, Table of tau(n);n=1 to 30.
Jean-Pierre Serre, An interpretation of some congruences concerning Ramanujan's tau function, Séminaire Delange-Pisot-Poitou. Théorie des nombres, tome 9, no 1 (1967-1968), exp. no 14, pp. 1-17.
Jean-Pierre Serre, An interpretation of some congruences concerning Ramanujan's Tau function, 1997.
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Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
David A. Steffen, Les Coefficients de Fourier de la forme modulaire: La fonction de Ramanujan tau(n), 1998.
William A. Stein, Database.
Zhi-Wei Sun, Questions on the reciprocals of the values of the tau function, Question 484763 at MathOverflow, December 25, 2024.
Zhi-Wei Sun, Pythagorean triples and Ramanujan's tau function, Question 484936 at MathOverflow, December 28, 2024.
Zhi-Wei Sun, Additive combinatorics for Ramanujan's tau function, Question 485138 at MathOverflow, January 1, 2025.
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
F. Van der Blij, The function tau (n) of S. Ramanujan (an expository lecture), Math. Student, Vol. 18, No. 3 (1950), pp. 83-99; entire issues 3 and 4.
Jan Vonk, Overconvergent modular forms and their explicit arithmetic, Bulletin of the American Mathematical Society 58.3 (2021): 313-356.
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Eric Weisstein's World of Mathematics, Tau Function.
Kenneth S. Williams, Historical remark on Ramanujan's tau function, Amer. Math. Monthly, 122 (2015), 30-35; author's copy.
Index entries for "core" sequences.
Index entries for expansions of Product_{k >= 1} (1-x^k)^m.
Index entries for sequences related to Chebyshev polynomials.

Cf. A076847 (tau(prime)), A278577 (prime powers), A037955, A027364, A037945, A037946, A037947, A008408 (Leech).
For a(n) mod N for various values of N see A046694, A098108, A126812-...
Cf. A006352, A099059, A099060, A262339, A292781.
For primes p such that tau(p) == -1 (mod 23) see A106867.
Cf. A010816, A004009. A013973.
Cf. A126832(n) = a(n) mod 5.

using Nemo
function DedekindEta(len, r)
    R, z = PolynomialRing(ZZ, "z")
    e = eta_qexp(r, len, z)
    [coeff(e, j) for j in 0:len - 1] end
RamanujanTauList(len) = DedekindEta(len, 24)
RamanujanTauList(28) |> println # Peter Luschny, Mar 09 2018

M12:=ModularForms(Gamma0(1),12); t1:=Basis(M12)[2]; PowerSeries(t1[1],100); Coefficients($1);

Basis( CuspForms( Gamma1(1), 12), 100)[1]; /* Michael Somos, May 27 2014 */

M := 50; t1 := series(x*mul((1-x^k)^24,k=1..M),x,M); A000594 := n-> coeff(t1,x,n);

CoefficientList[ Take[ Expand[ Product[ (1 - x^k)^24, {k, 1, 30} ]], 30], x] (* Or *)
(* first do *) Needs["NumberTheory`Ramanujan`"] (* then *) Table[ RamanujanTau[n], {n, 30}] (* Dean Hickerson, Jan 03 2003 *)
max = 28; g[k_] := -BernoulliB[k]/(2k) + Sum[ DivisorSigma[k - 1, n - 1]*q^(n - 1), {n, 2, max + 1}]; CoefficientList[ Series[ 8000*g[4]^3 - 147*g[6]^2, {q, 0, max}], q] // Rest (* Jean-François Alcover, Oct 10 2012, from modular forms *)
RamanujanTau[Range[40]] (* The function RamanujanTau is now part of Mathematica's core language so there is no longer any need to load NumberTheory`Ramanujan` before using it *) (* Harvey P. Dale, Oct 12 2012 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^24, {q, 0, n}]; (* Michael Somos, May 27 2014 *)
a[ n_] := With[{t = Log[q] / (2 Pi I)}, SeriesCoefficient[ Series[ DedekindEta[t]^24, {q, 0, n}], {q, 0, n}]]; (* Michael Somos, May 27 2014 *)

{a(n) = if( n<1, 0, polcoeff( x * eta(x + x * O(x^n))^24, n))};

{a(n) = if( n<1, 0, polcoeff( x * (sum( i=1, (sqrtint( 8*n - 7) + 1) \ 2,(-1)^i * (2*i - 1) * x^((i^2 - i)/2), O(x^n)))^8, n))};

taup(p,e)={
    if(e==1,
        (65*sigma(p,11)+691*sigma(p,5)-691*252*sum(k=1,p-1,sigma(k,5)*sigma(p-k,5)))/756
    ,
        my(t=taup(p,1));
        sum(j=0,e\2,
            (-1)^j*binomial(e-j,e-2*j)*p^(11*j)*t^(e-2*j)
        )
    )
};
a(n)=my(f=factor(n));prod(i=1,#f[,1],taup(f[i,1],f[i,2]));
\\ Charles R Greathouse IV, Apr 22 2013

\\ compute terms individually (Douglas Niebur, Ill. J. Math., 19, 1975):
a(n) = n^4*sigma(n) - 24*sum(k=1, n-1, (35*k^4-52*k^3*n+18*k^2*n^2)*sigma(k)*sigma(n-k));
vector(33, n, a(n)) \\ Joerg Arndt, Sep 06 2015

a(n)=ramanujantau(n) \\ Charles R Greathouse IV, May 27 2016

from sympy import divisor_sigma
def A000594(n): return n**4*divisor_sigma(n)-24*((m:=n+1>>1)**2*(0 if n&1 else (m*(35*m - 52*n) + 18*n**2)*divisor_sigma(m)**2)+sum((i*(i*(i*(70*i - 140*n) + 90*n**2) - 20*n**3) + n**4)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))) # Chai Wah Wu, Nov 08 2022

def s(n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0}
  s
end
def A000594(n)
  ary = [1]
  a = [0] + (1..n - 1).map{|i| s(i)}
  (1..n - 1).each{|i| ary << (1..i).inject(0){|s, j| s - 24 * a[j] * ary[-j]} / i}
  ary
end
p A000594(100) # Seiichi Manyama, Mar 26 2017

def A000594(n)
  ary = [0, 1]
  (2..n).each{|i|
    s, t, u = 0, 1, 0
    (1..n).each{|j|
      t += 9 * j
      u += j
      break if i <= u
      s += (-1) ** (j % 2 + 1) * (2 * j + 1) * (i - t) * ary[-u]
    }
    ary << s / (i - 1)
  }
  ary[1..-1]
end
p A000594(100) # Seiichi Manyama, Nov 25 2017

CuspForms( Gamma1(1), 12, prec=100).0; # Michael Somos, May 28 2013

list(delta_qexp(100))[1:] # faster Peter Luschny, May 16 2016

G.f.: x * Product_{k>=1} (1 - x^k)^24 = x*A(x)^8, with the g.f. of A010816.
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^12 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 04 2011
abs(a(n)) = O(n^(11/2 + epsilon)), abs(a(p)) <= 2 p^(11/2) if p is prime. These were conjectured by Ramanujan and proved by Deligne.
Zagier says: The proof of these formulas, if written out from scratch, has been estimated at 2000 pages; in his book Manin cites this as a probable record for the ratio: "length of proof:length of statement" in the whole of mathematics.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w * (u + 48*v + 4096*w) - v^3. - Michael Somos, Jul 19 2004
G.f. A(q) satisfies q * d log(A(q))/dq = A006352(q). - Michael Somos, Dec 09 2013
a(2*n) = A099060(n). a(2*n + 1) = A099059(n). - Michael Somos, Apr 17 2015
a(n) = tau(n) (with tau(0) = 0): tau(m)*tau(n) = Sum_{d| gcd(m,n)} d^11*tau(m*n/d^2), for positive integers m and n. If gcd(m,n) = 1 this gives the multiplicativity of tau. See a comment above with the Koecher-Krieg reference, p. 212, eq. (5). - Wolfdieter Lang, Jan 21 2016
Dirichlet series as product: Sum_{n >= 1} a(n)/n^s = Product_{n >= 1} 1/(1 - a(prime(n))/prime(n)^s  + prime(n)^(11-2*s)). See the Mordell link, eq. (2). - Wolfdieter Lang, May 06 2016. See also Hardy, p. 164, eqs. (10.3.1) and (10.3.8). - Wolfdieter Lang, Jan 27 2017
a(n) is multiplicative with a(prime(n)^k) = sqrt(prime(n)^(11))^k*S(k, a(n) / sqrt(prime(n)^(11))), with the Chebyshev S polynomials (A049310), for n >= 1 and k >= 2, and A076847(n) = a(prime(n)). See A076847 for alpha multiplicativity and examples. - Wolfdieter Lang, May 17 2016. See also Hardy, p. 164, eq. (10.3.6) rewritten in terms of S. - Wolfdieter Lang, Jan 27 2017
G.f. eta(z)^24 (with q = exp(2*Pi*i*z)) also (E_4(q)^3 - E_6(q)^2) / 1728. See the Hardy reference, p. 166, eq. (10.5.3), with Q = E_4 and R = E_6, given in A004009 and A013973, respectively. - Wolfdieter Lang, Jan 30 2017
a(n) (mod 5) == A126832(n).
a(1) = 1, a(n) = -(24/(n-1))*Sum_{k=1..n-1} A000203(k)*a(n-k) for n > 1. - Seiichi Manyama, Mar 26 2017
G.f.: x*exp(-24*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Euler Transform of [-24, -24, -24, -24, ...]. - Simon Plouffe, Jun 21 2018
a(n) = n^4*sigma(n)-24*Sum_{k=1..n-1} (35*k^4-52*k^3*n+18*k^2*n^2)*sigma(k)*sigma(n-k). [See Douglas Niebur link]. - Wesley Ivan Hurt, Jul 22 2025

A002054 Binomial coefficient C(2n+1, n-1).

Original entry on oeis.org

1, 5, 21, 84, 330, 1287, 5005, 19448, 75582, 293930, 1144066, 4457400, 17383860, 67863915, 265182525, 1037158320, 4059928950, 15905368710, 62359143990, 244662670200, 960566918220, 3773655750150, 14833897694226, 58343356817424, 229591913401900

Offset: 1

N. J. A. Sloane

a(n) = number of permutations in S_{n+2} containing exactly one 312 pattern. E.g., S_3 has a_1 = 1 permutations containing exactly one 312 pattern, and S_4 has a_2 = 5 permutations containing exactly one 312 pattern, namely 1423, 2413, 3124, 3142, and 4231. This comment is also true if 312 is replaced by any of 132, 213, or 231 (but not 123 or 321, for which see A003517). [Comment revised by N. J. A. Sloane, Nov 26 2022]
Number of valleys in all Dyck paths of semilength n+1. Example: a(2)=5 because UD*UD*UD, UD*UUDD, UUDD*UD, UUD*UDD, UUUDDD, where U=(1,1), D=(1,-1) and the valleys are shown by *. - Emeric Deutsch, Dec 05 2003
Number of UU's (double rises) in all Dyck paths of semilength n+1. Example: a(2)=5 because UDUDUD, UDU*UDD, U*UDDUD, U*UDUDD, U*U*UDDD, the double rises being shown by *. - Emeric Deutsch, Dec 05 2003
Number of peaks at level higher than one (high peaks) in all Dyck paths of semilength n+1. Example: a(2)=5 because UDUDUD, UDUU*DD, UU*DDUD, UU*DU*DD, UUU*DDD, the high peaks being shown by *. - Emeric Deutsch, Dec 05 2003
Number of diagonal dissections of a convex (n+3)-gon into n regions. Number of standard tableaux of shape (n,n,1) (see Stanley reference). - Emeric Deutsch, May 20 2004
Number of dissections of a convex (n+3)-gon by noncrossing diagonals into several regions, exactly n-1 of which are triangular. Example: a(2)=5 because the convex pentagon ABCDE is dissected by any of the diagonals AC, BD, CE, DA, EB into regions containing exactly 1 triangle. - Emeric Deutsch, May 31 2004
Number of jumps in all full binary trees with n+1 internal nodes. In the preorder traversal of a full binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump. - Emeric Deutsch, Jan 18 2007
a(n) is the total number of nonempty Dyck subpaths in all Dyck paths (A000108) of semilength n. For example, the Dyck path UUDUUDDD has Dyck subpaths stretching over positions 1-8 (the entire path), 2-3, 2-7, 4-7, 5-6 and so contributes 5 to a(4). - David Callan, Jul 25 2008
a(n+1) is the total number of ascents in the set of all n-permutations avoiding the pattern 132. For example, a(2) = 5 because there are 5 ascents in the set 123, 213, 231, 312, 321. - Cheyne Homberger, Oct 25 2013
Number of increasing tableaux of shape (n+1,n+1) with largest entry 2n+1. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. Example: a(2) = 5 counts the five tableaux (124)(235), (123)(245), (124)(345), (134)(245), (123)(245). - Oliver Pechenik, May 02 2014
a(n) is the number of noncrossing partitions of 2n+1 into n-1 blocks of size 2 and 1 block of size 3. - Oliver Pechenik, May 02 2014
Number of paths in the half-plane x>=0, from (0,0) to (2n+1,3), and consisting of steps U=(1,1) and D=(1,-1). For example, for n=2, we have the 5 paths: UUUUD, UUUDU, UUDUU, UDUUU, DUUUU. - José Luis Ramírez Ramírez, Apr 19 2015
From Gus Wiseman, Aug 20 2021: (Start)
Also the number of binary numbers with 2n+2 digits and with two more 0's than 1's. For example, the a(2) = 5 binary numbers are: 100001, 100010, 100100, 101000, 110000, with decimal values 33, 34, 36, 40, 48. Allowing first digit 0 gives A001791, ranked by A345910/A345912.
Also the number of integer compositions of 2n+2 with alternating sum -2, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. For example, the a(3) = 21 compositions are:
  (35)  (152)  (1124)  (11141)  (111113)
        (251)  (1223)  (12131)  (111212)
               (1322)  (13121)  (111311)
               (1421)  (14111)  (121112)
               (2114)           (121211)
               (2213)           (131111)
               (2312)
               (2411)
The following pertain to these compositions:
- The unordered version is A344741.
- Ranked by A345924 (reverse: A345923).
- A345197 counts compositions by length and alternating sum.
- A345925 ranks compositions with alternating sum 2 (reverse: A345922).
(End)

			G.f. = x + 5*x^2 + 21*x^3 + 84*x^4 + 330*x^5 + 1287*x^6 + 5005*x^7 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
George Grätzer, General Lattice Theory. Birkhauser, Basel, 1998, 2nd edition, p. 474, line -3.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 1..100 computed by T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
F. R. Bernhart and N. J. A. Sloane, Emails, April-May 1994.
Jean-Luc Baril and Sergey Kirgizov, The pure descent statistic on permutations, Discrete Mathematics, Vol. 340, No. 10 (2017), pp. 2550-2558; preprint, 2017.
Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Dyck Paths with catastrophes modulo the positions of a given pattern, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.
David Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
Arthur Cayley, On the partitions of a polygon, Proc. London Math. Soc., Vol. 22 (1891), pp. 237-262 = Collected Mathematical Papers, Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.
Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020.
Colin Defant, Proofs of Conjectures about Pattern-Avoiding Linear Extensions, arXiv:1905.02309 [math.CO], 2019.
Emeric Deutsch, Dyck path enumeration, Discrete Math., Vol. 204, No. 1-3 (1999), pp. 167-202.
Gennady Eremin, Dyck Numbers, IV. Nested patterns in OEIS A036991, arXiv:2306.10318 [math.CO], 2023. See (5) p. 7.
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, J. Int. Seq., Vol. 17 (2014), Article 14.1.5; arXiv preprint, arXiv:1203.6792 [math.CO], 2012.
Xiaoyu He, Emily Huang, Ihyun Nam and Rishubh Thaper, Shuffle Squares and Reverse Shuffle Squares, arXiv:2109.12455 [math.CO], 2021.
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
Milan Janjic, Two Enumerative Functions.
Werner Krandick, Trees and jumps and real roots, J. Computational and Applied Math., Vol. 162, No. 1 (2004), pp. 51-55.
Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.5.
Toufik Mansour and Alek Vainshtein, Counting occurrences of 123 in a permutation, arXiv:math/0105073 [math.CO], 2001.
Henri Mühle, Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices, arXiv:1509.06942 [math.CO], 2015.
Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Vol. 15 (2012), Article 12.3.3.
John Noonan and Doron Zeilberger, The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns, arXiv:math/9808080 [math.CO], 1998.
Oliver Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, arXiv:1209.1355 [math.CO], 2012-2014.
Oliver Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, J. Combin. Theory A, Vol. 125 (2014), pp. 357-378.
Karol Penson, Hausdorff moment problems for combinatorial numbers: heuristics via Meijer G-functions, Nov. 2022.
Ronald C. Read, On general dissections of a polygon, Aequationes Mathematicae, Vol. 18, No. 1-2 (1978), pp. 370-388; Preprint, 1974.
Mark Shattuck, Enumeration of non-crossing partitions according to subwords with repeated letters, arXiv:2303.06300 [math.CO], 2023.
Richard P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, Vol. 76 , No. 1 (1996), pp. 175-177.
Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).
A. Vogt, Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation, arXiv:1108.2993 [hep-ph], 2011.
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.

Diagonal 4 of triangle A100257. Also a diagonal of A033282.
Equals (1/2) A024483(n+2). Bisection of A037951 and A037955.
Cf. A001263.
Column k=1 of A263771.
Counts terms of A031445 with 2n+2 digits in binary.
Cf. A000097, A000346, A000984, A001622, A001700, A007318, A008549, A031444, A058622, A097805, A116406, A138364, A163493, A202736.
Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002694 (m = 4), A003516 (m = 5), A002696 (m = 6), A030053 - A030056, A004310 - A004318.

List([1..25],n->Binomial(2*n+1,n-1)); # Muniru A Asiru, Aug 09 2018

[Binomial(2*n+1, n-1): n in [1..30]]; // Vincenzo Librandi, Apr 20 2015

with(combstruct): seq((count(Composition(2*n+2), size=n)), n=1..24); # Zerinvary Lajos, May 03 2007

CoefficientList[Series[8/(((Sqrt[1-4x] +1)^3)*Sqrt[1-4x]), {x,0,22}], x] (* Robert G. Wilson v, Aug 08 2011 *)
a[ n_]:= Binomial[2 n + 1, n - 1]; (* Michael Somos, Apr 25 2014 *)

PARI
```
{a(n) = binomial( 2*n+1, n-1)};
```

from _future_ import division
A002054_list, b = [], 1
for n in range(1,10**3):
    A002054_list.append(b)
    b = b*(2*n+2)*(2*n+3)//(n*(n+3)) # Chai Wah Wu, Jan 26 2016

[binomial(2*n+1, n-1) for n in (1..25)] # G. C. Greubel, Mar 22 2019

a(n) = Sum_{j=0..n-1} binomial(2*j, j) * binomial(2*n - 2*j, n-j-1)/(j+1). - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
G.f.: z*C^4/(2-C), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. - Emeric Deutsch, Jul 05 2003
From Wolfdieter Lang, Jan 09 2004: (Start)
a(n) = binomial(2*n+1, n-1) = n*C(n+1)/2, C(n)=A000108(n) (Catalan).
G.f.: (1 - 2*x - (1-3*x)*c(x))/(x*(1-4*x)) with g.f. c(x) of A000108. (End)
G.f.: z*C(z)^3/(1-2*z*C(z)), where C(z) is the g.f. of Catalan numbers. - José Luis Ramírez Ramírez, Apr 19 2015
G.f.: 2F1(5/2, 2; 4; 4*x). - R. J. Mathar, Aug 09 2015
D-finite with recurrence: a(n+1) = a(n)*(2*n+3)*(2*n+2)/(n*(n+3)). - Chai Wah Wu, Jan 26 2016
From Ilya Gutkovskiy, Aug 30 2016: (Start)
E.g.f.: (BesselI(0,2*x) + (1 - 1/x)*BesselI(1,2*x))*exp(2*x).
a(n) ~ 2^(2*n+1)/sqrt(Pi*n). (End)
a(n) = (1/(n+1))*Sum_{i=0..n-1} (n+1-i)*binomial(2n+2,i), n >= 1. - Taras Goy, Aug 09 2018
G.f.: (x - 1 + (1 - 3*x)/sqrt(1 - 4*x))/(2*x^2). - Michael Somos, Jul 28 2021
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=1} 1/a(n) = 5/3 - 2*Pi/(9*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 52*log(phi)/(5*sqrt(5)) - 7/5, where phi is the golden ratio (A001622). (End)
a(n) = A001405(2*n+1) - A000108(n+1), n >= 1 (from Eremin link, page 7). - Gennady Eremin, Sep 05 2023
G.f.: x/(1 - 4*x)^2 * c(-x/(1 - 4*x))^3, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Feb 03 2024
From Peter Bala, Oct 13 2024: (Start)
a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * sqrt(x)*(x - 3)/sqrt(4 - x) (see Penson).
G.f. x*/sqrt(1 - 4*x) * c(x)^3. (End)

A061554 Square table read by antidiagonals: a(n,k) = binomial(n+k, floor(k/2)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 4, 4, 1, 1, 10, 10, 5, 5, 1, 1, 20, 15, 15, 6, 6, 1, 1, 35, 35, 21, 21, 7, 7, 1, 1, 70, 56, 56, 28, 28, 8, 8, 1, 1, 126, 126, 84, 84, 36, 36, 9, 9, 1, 1, 252, 210, 210, 120, 120, 45, 45, 10, 10, 1, 1, 462, 462, 330, 330, 165, 165, 55, 55, 11, 11, 1, 1

Offset: 0

Henry Bottomley, May 17 2001

Equivalently, a triangle read by rows, where the rows are obtained by sorting the elements of the rows of Pascal's triangle (A007318) into descending order. - Philippe Deléham, May 21 2005
Equivalently, as a triangle read by rows, this is T(n,k)=binomial(n,floor((n-k)/2)); column k then has e.g.f. Bessel_I(k,2x)+Bessel_I(k+1,2x). - Paul Barry, Feb 28 2006
Antidiagonal sums are A037952(n+1) = C(n+1,[n/2]). Matrix inverse is the row reversal of triangle A066170. Eigensequence is A125094(n) = Sum_{k=0..n-1} A125093(n-1,k)*A125094(k). - Paul D. Hanna, Nov 20 2006
Riordan array (1/(1-x-x^2*c(x^2)),x*c(x^2)); where c(x)=g.f.for Catalan numbers A000108. - Philippe Deléham, Mar 17 2007
Triangle T(n,k), 0<=k<=n, read by rows given by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+T(n-1,k+1) for k>=1. - Philippe Deléham, Mar 27 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1. Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
T(n,k) is the number of paths from (0,k) to some (n,m) which never dip below y=0, touch y=0 at least once and are made up only of the steps (1,1) and (1,-1). This can be proved using the recurrence supplied by Deléham. - Gerald McGarvey, Oct 15 2008
Triangle read by rows = partial sums of A053121 terms starting from the right. - Gary W. Adamson, Oct 24 2008
As a subset of the "family of triangles" (Deleham comment of Sep 25 2007), beginning with a signed variant of A061554, M = (-1,0) =  (1; -1, 1; 2, -1, 1; -3, 3, -1, 1; ...) successive binomial transforms of M yield (0,1) - A089942; (1,2) - A039599; (2,3) - A124733; (3,4) - A124574; (4,5) - A126331; ... such that the binomial transform of the triangle generated from (n,n+1) = the triangle generated from (n+1,n+2). Similarly, another subset beginning with A053121 - (0,0), and taking successive binomial transforms yields (1,1) - A064189; (2,2) - A039598; (3,3) - A091965, ... By rows, the triangle generated from (n,n) can be obtained by taking pairwise sums from the (n-1,n) triangle starting from the right. For example, row 2 of (1,2) - A039599 = (2, 3, 1); and taking pairwise sums from the right we obtain (5, 4, 1) = row 2 of (2,2) - A039598. - Gary W. Adamson, Aug 04 2011
The triangle by rows (n) with alternating signs (+-+...) from the top as a set of simultaneous equations solves for diagonal lengths of odd N (N = 2n+1) regular polygons. The constants in each case are powers of c = 2*cos(2*Pi/N). By way of example, the first 3 rows relate to the heptagon and the simultaneous equations are (1,0,0) = 1; (-1,1,0) = c = 1.24697...; and (2,-1,1) = c^2. The answers are 1, 2.24697..., and 1.801...; the 3 distinct diagonal lengths of the heptagon with edge = 1. - Gary W. Adamson, Sep 07 2011

			The array starts:
   1, 1,  2,  3,  6, 10,  20,  35,   70,  126, ...
   1, 1,  3,  4, 10, 15,  35,  56,  126,  210, ...
   1, 1,  4,  5, 15, 21,  56,  84,  210,  330, ...
   1, 1,  5,  6, 21, 28,  84, 120,  330,  495, ...
   1, 1,  6,  7, 28, 36, 120, 165,  495,  715, ...
   1, 1,  7,  8, 36, 45, 165, 220,  715, 1001, ...
   1, 1,  8,  9, 45, 55, 220, 286, 1001, 1365, ...
   1, 1,  9, 10, 55, 66, 286, 364, 1365, 1820, ...
   1, 1, 10, 11, 66, 78, 364, 455, 1820, 2380, ...
   1, 1, 11, 12, 78, 91, 455, 560, 2380, 3060, ...
Triangle (antidiagonal) version begins:
    1;
    1,   1;
    2,   1,   1;
    3,   3,   1,   1;
    6,   4,   4,   1,   1;
   10,  10,   5,   5,   1,   1;
   20,  15,  15,   6,   6,   1,  1;
   35,  35,  21,  21,   7,   7,  1,  1;
   70,  56,  56,  28,  28,   8,  8,  1,  1;
  126, 126,  84,  84,  36,  36,  9,  9,  1,  1;
  252, 210, 210, 120, 120,  45, 45, 10, 10,  1, 1;
  462, 462, 330, 330, 165, 165, 55, 55, 11, 11, 1, 1; ...
Matrix inverse begins:
   1;
  -1,  1;
  -1, -1,   1;
   1, -2,  -1,   1;
   1,  2,  -3,  -1,  1;
  -1,  3,   3,  -4, -1,  1;
  -1, -3,   6,   4, -5, -1,  1;
   1, -4,  -6,  10,  5, -6, -1,  1;
   1,  4, -10, -10, 15,  6, -7, -1, 1; ...
From _Paul Barry_, May 21 2009: (Start)
Production matrix is
  1, 1,
  1, 0, 1,
  0, 1, 0, 1,
  0, 0, 1, 0, 1,
  0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 1, 0, 1,
  0, 0, 0, 0, 0, 1, 0, 1 (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Reed Acton, T. Kyle Petersen, Blake Shirman, and Bridget Eileen Tenner, The clairvoyant maître d', arXiv:2401.11680 [math.CO], 2024. See p. 15.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Yidong Sun and Luping Ma, Minors of a class of Riordan arrays related to weighted partial Motzkin paths. Eur. J. Comb. 39, 157-169 (2014), Table 2.2.

Rows are A001405, A037952, A037955, A037951, A037956, A037953, A037957 etc. Columns are truncated pairs of A000012, A000027, A000217, A000292, A000332, A000389, A000579, etc. Main diagonal is alternate values of A051036.

Cf. A007318, A107430, A125094, A037952, A066170.

T := proc(n, k) option remember;
if n = k then 1 elif k < 0 or n < 0 or k > n then 0
elif k = 0 then T(n-1, 0) + T(n-1, 1) else T(n-1, k-1) + T(n-1, k+1) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # Peter Luschny, May 25 2021

t[n_, k_] = Binomial[n, Floor[(n+1)/2 - (-1)^(n-k)*(k+1)/2]]; Flatten[Table[t[n, k], {n, 0, 11}, {k, 0, n}]] (* Jean-François Alcover, May 31 2011 *)

T(n,k)=binomial(n,(n+1)\2-(-1)^(n-k)*((k+1)\2))

As a triangle: T(n,k) = binomial(n,m) where m = floor((n+1)/2 - (-1)^(n-k)*(k+1)/2).
a(0, k) = binomial(k, floor(k/2)) = A001405(k); for n>0 T(n, k) = T(n+1, k-2) + T(n-1, k).
n-th row = M^n * V, where M = the infinite tridiagonal matrix with all 1's in the super and subdiagonals and (1,0,0,0,...) in the main diagonal. V = the infinite vector [1,0,0,0,...]. Example: (3,3,1,1,0,0,0,...) = M^3 * V. - Gary W. Adamson, Nov 04 2006
Sum_{k=0..n} T(m,k)*T(n,k) = T(m+n,0) = A001405(m+n). - Philippe Deléham, Feb 26 2007
Sum_{k=0..n} T(n,k)=2^n. - Philippe Deléham, Mar 27 2007
Sum_{k=0..n} T(n,k)*x^k = A127361(n), A126869(n), A001405(n), A000079(n), A127358(n), A127359(n), A127360(n) for x = -2, -1, 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Dec 04 2009

Entry revised by N. J. A. Sloane, Nov 22 2006

A034869 Right half of Pascal's triangle.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 6, 4, 1, 10, 5, 1, 20, 15, 6, 1, 35, 21, 7, 1, 70, 56, 28, 8, 1, 126, 84, 36, 9, 1, 252, 210, 120, 45, 10, 1, 462, 330, 165, 55, 11, 1, 924, 792, 495, 220, 66, 12, 1, 1716, 1287, 715, 286, 78, 13, 1, 3432, 3003, 2002, 1001, 364, 91, 14, 1

Offset: 0

N. J. A. Sloane

From R. J. Mathar, May 13 2006: (Start)
Also flattened table of the expansion coefficients of x^n in Chebyshev Polynomials T_k(x) of the first kind:
x^n is 2^(1-n) multiplied by the sum of floor(1+n/2) terms using only terms T_k(x) with even k if n even, only terms T_k(x) with odd k if n is odd and halving the coefficient a(..) in front of any T_0(x):
x^0=2^(1-0) a(0)/2 T_0(x)
x^1=2^(1-1) a(1) T_1(x)
x^2=2^(1-2) [a(2)/2 T_0(x)+a(3) T_2(x)]
x^3=2^(1-3) [a(4) T_1(x)+a(5) T_3(x)]
x^4=2^(1-4) [a(6)/2 T_0(x)+a(7) T_2(x) +a(8) T_4(x)]
x^5=2^(1-5) [a(9) T_1(x)+a(10) T_3(x) +a(11) T_5(x)]
x^6=2^(1-6) [a(12)/2 T_0(x)+a(13) T_2(x) +a(14) T_4(x) +a(15) T_6(x)]
x^7=2^(1-7) [a(16) T_1(x)+a(17) T_3(x) +a(18) T_5(x) +a(19) T_7(x)]" (End)
T(n,k) = A034868(n,floor(n/2)-k), k = 0..floor(n/2). - Reinhard Zumkeller, Jul 27 2012
Rows are binomial(r-1,(2r+1-(-1)^r)\4 -n ) where r is the row and n is the term. Columns are binomial(2m+c-3,m-1) where c is the column and m is the term. - Anthony Browne, May 17 2016

			The table starts:
  1
  1
  2 1
  3 1
  6 4 1
  ...

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Cf. A007318, A008619 (row lengths).
Cf. A110654.
Cf. A034868 (left half), A014413, A014462, A027306 (row sums).
Columns k=0-1-2-3-4 give: A001405, A037955, A037956, A037957, A037958.

a034869 n k = a034869_tabf !! n !! k
a034869_row n = a034869_tabf !! n
a034869_tabf = [1] : f 0 [1] where
   f 0 us'@(_:us) = ys : f 1 ys where
                    ys = zipWith (+) us' (us ++ [0])
   f 1 vs@(v:_) = ys : f 0 ys where
                  ys = zipWith (+) (vs ++ [0]) ([v] ++ vs)
-- Reinhard Zumkeller, improved Dec 21 2015, Jul 27 2012

for n from 0 to 60 do for j from n mod 2 to n by 2 do print( binomial(n,(n-j)/2) ); od; od; # R. J. Mathar, May 13 2006
# Second program:
egf:= k-> BesselI(2*k, 2*x) + BesselI(2*k+1, 2*x):
A034869:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(print(seq(A034869(n, k), k=0..iquo(n, 2))), n=0..14); # Mélika Tebni, Sep 05 2024

Table[Binomial[n, k], {n, 0, 14}, {k, Ceiling[n/2], n}] // Flatten (* Michael De Vlieger, May 19 2016 *)

for(n=0, 14, for(k=ceil(n/2), n, print1(binomial(n, k),", ");); print();) \\ Indranil Ghosh, Mar 31 2017

import math
from sympy import binomial
for n in range(15):
    print([binomial(n, k) for k in range(math.ceil(n/2), n + 1)]) # Indranil Ghosh, Mar 31 2017

E.g.f. of column k: BesselI(2*k,2*x) + BesselI(2*k+1,2*x). - Mélika Tebni, Sep 05 2024

Keyword fixed and example added by Franklin T. Adams-Watters, May 27 2010

A101491 Triangle T(n,k), read by rows: number of Knödel walks starting at 0, ending at k, with n steps.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 1, 3, 1, 1, 5, 4, 4, 1, 1, 5, 10, 5, 5, 1, 1, 15, 15, 15, 6, 6, 1, 1, 20, 35, 21, 21, 7, 7, 1, 1, 50, 56, 56, 28, 28, 8, 8, 1, 1, 76, 126, 84, 84, 36, 36, 9, 9, 1, 1, 176, 210, 210, 120, 120, 45, 45, 10, 10, 1, 1, 286, 462, 330, 330, 165, 165, 55, 55, 11, 11, 1, 1

Offset: 0

Ralf Stephan, Jan 21 2005

			Triangle begins:
  1,
  0,1,
  2,1,1,
  1,3,1,1,
  5,4,4,1,1,
  5,10,5,5,1,1,
  15,15,15,6,6,1,1,
  20,35,21,21,7,7,1,1,
  50,56,56,28,28,8,8,1,1,
  76,126,84,84,36,36,9,9,1,1,
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
Helmut Prodinger, The Kernel Method: a collection of examples, Séminaire Lotharingien de Combinatoire, B50f (2004), 19 pp.

Left-hand columns include A086905, A037952, A037955, A037951, A037956, A037953, A037957, A037954, A037958.

A101491[n_, k_] := If[k == 0, Sum[(-1)^(n - i)*Binomial[i, BitShiftRight[i]], {i, 0, n}], Binomial[n, BitShiftRight[n - k]]];
Table[A101491[n, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Jan 17 2025 *)

T(n, k) = if (k==0, sum(i=0, n, (-1)^(n-i)*binomial(i, i\2)), binomial(n, (n-k)\2));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print();); \\ Michel Marcus, Dec 04 2016

G.f.: r(z)/(z*(1+z)*(1-r(z)))*(1+x*z*r(z))/(1-x*r(z)), with r(z) = (1-sqrt(1-4*z^2))/(2*z). Then the g.f. of the k-th column is r(z)^(k+1)/(z*(1-r(z))).

T(n, k) = Sum_{i=0..n} (-1)^(n-i)*C(i, floor(i/2)) for k=0, otherwise T(n, k) = C(n, floor((n-k)/2)).

A005775 Number of compact-rooted directed animals of size n having 3 source points.

Original entry on oeis.org

1, 4, 14, 45, 140, 427, 1288, 3858, 11505, 34210, 101530, 300950, 891345, 2638650, 7809000, 23107488, 68375547, 202336092, 598817490, 1772479905, 5247421410, 15538054455, 46019183840, 136325212750, 403933918375, 1197131976846, 3548715207534, 10521965227669

Offset: 3

Simon Plouffe

Binomial transform of A037955. - Paul Barry, Dec 28 2006
Apparently, the number of Dyck paths of semilength n that contain at least one UUU but avoid UUU's starting above level 0. - David Scambler, Jul 02 2013
a(n) = number of paths in the half-plane x >= 0 from (0,0) to (n-1,2) or (n-1,-3), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=5, we have the 14 paths: HHUU, UUHH, UHHU, HUUH, HUHU, UHUH, UDUU, UUDU, UUUD, DUUU, DDDH, HDDD, DHDD, DDHD. - José Luis Ramírez Ramírez, Apr 19 2015

			G.f. = x^3 + 4*x^4 + 14*x^5 + 45*x^6 + 140*x^7 + 427*x^8 + 1288*x^9 + 3858*x^10 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 3..1000
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
D. Gouyou-Beauchamps and G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), no. 3, 334-357.

Cf. A005773.
k=2 column of array in A038622.
Cf. A005774, A066822.

a005775 = flip a038622 2 . (subtract 1)  -- Reinhard Zumkeller, Feb 26 2013

seq(simplify(GegenbauerC(n-4,-n+1,-1/2) + GegenbauerC(n-3,-n+1,-1/2)),n=3..28); # Peter Luschny, May 12 2016

nmax = 28; t[n_ /; n > 0, k_ /; k >= 1] := t[n, k] = t[n-1, k-1] + t[n-1, k] + t[n-1, k+1]; t[0, 0] = 1; t[0, ] = 0; t[?Negative, ?Negative] = 0; t[n, 0] := 2*t[n-1, 0] + t[n-1, 1]; a[n_] := t[n-1, 2]; Table[a[n], {n, 3, nmax} ] (* Jean-François Alcover, Jul 03 2013, from A038622 *)

{a(n) = polcoeff( (x^2 + x - 1 + (x^2 - 3*x + 1) * sqrt((1 + x) / (1 - 3*x) + x^3 * O(x^n))) / (2*x^2), n)};

{a(n) = n--; sum(k=0, n, binomial(n, k) * binomial(k, k\2 -1))}; /* Michael Somos, May 12 2016 */

D-finite with recurrence (n+2)*(n-3)*a(n) = 2*n*(n-1)*a(n-1) + 3*(n-1)*(n-2)*a(n-2), a(2)=0, a(3)=1. - Michael Somos, Feb 02 2002
G.f.: (x^2 + x - 1 +(x^2 - 3*x + 1)*sqrt((1+x)/(1-3*x)))/(2*x^2).
From Paul Barry, Dec 28 2006: (Start)
E.g.f.: exp(x)*(Bessel_I(2,2*x) + Bessel_I(3,2*x));
a(n+1) = Sum_{k=0..n} C(n,k)*C(k,floor(k/2)-1). (End)
a(n) ~ 3^(n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 25 2014
G.f.: (z^3*M(z)^2+z^4*M(z)^3)/(1-z-2*z^2*M(z)), where M(z) is the g.f. of Motzkin paths. - José Luis Ramírez Ramírez, Apr 19 2015
a(n) = GegenbauerC(n-4,-n+1,-1/2) + GegenbauerC(n-3,-n+1,-1/2). - Peter Luschny, May 12 2016
0 = a(n)*(+9*a(n+1) - 63*a(n+2) - 54*a(n+3) + 87*a(n+4) - 21*a(n+5))+ a(n+1)*(+21*a(n+1) + 79*a(n+2) + 13*a(n+3) - 118*a(n+4) + 35*a(n+5)) + a(n+2)*(-14*a(n+2) + 79*a(n+3) - 67*a(n+4) + 14*a(n+5)) + a(n+3)*(+6*a(n+3) + 19*a(n+4) - 11*a(n+5)) + a(n+4)*(+a(n+4) + a(n+5)) if n >= 0. - Michael Somos, May 12 2016
a(n) = A005773(n) - A001006(n) for n >= 3. - John Keith, Nov 20 2020

More terms from Randall L Rathbun, Jan 19 2002

Edited by Michael Somos, Feb 02 2002

A089940 Triangle read by rows: T(n,k)=binomial(n+k,floor((n-k)/2)).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 6, 5, 6, 1, 1, 10, 15, 7, 8, 1, 1, 20, 21, 28, 9, 10, 1, 1, 35, 56, 36, 45, 11, 12, 1, 1, 70, 84, 120, 55, 66, 13, 14, 1, 1, 126, 210, 165, 220, 78, 91, 15, 16, 1, 1, 252, 330, 495, 286, 364, 105, 120, 17, 18, 1, 1, 462, 792, 715, 1001, 455, 560, 136, 153

Offset: 0

Paul Barry, Nov 16 2003

Columns include A001405, A037955, A037956, A037957. Row sums are A089941.

A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 24, 12, 4, 6, 1, 120, 60, 20, 30, 5, 10, 1, 720, 360, 120, 180, 30, 60, 6, 90, 15, 20, 1, 5040, 2520, 840, 1260, 210, 420, 42, 630, 105, 140, 7, 210, 21, 35, 1, 40320, 20160, 6720, 10080, 1680, 3360, 336, 5040, 840, 1120, 56

Offset: 0

Tilman Piesk, Nov 04 2014

This triangle shows the same numbers in each row as A036038 and A078760 (the multinomial coefficients), but in this arrangement the multisets in column n correspond to the n-th integer partition in the infinite order defined by A194602.
Row lengths: A000041 (partition numbers), Row sums: A005651
Columns: 0: A000142 (factorials), 1: A001710, 2: A001715, 3: A133799, 4: A001720, 6: A001725, 10: A001730, 14: A049388
Last in row: end-2: A037955 after 1 term mismatch, end-1: A001405, end: A000012
The rightmost columns form the triangle A173333:
  n       0      1     2     4    6  10  14  21      (A000041(1,2,3...)-1)
m
1         1
2         2      1
3         6      3     1
4        24     12     4     1
5       120     60    20     5    1
6       720    360   120    30    6   1
7      5040   2520   840   210   42   7   1
8     40320  20160  6720  1680  336  56   8   1
A249620 shows the number of partitions of the same multisets. A187783 shows the number of permutations of special multisets.

			Triangle begins:
  n     0    1    2    3   4   5  6   7   8   9 10
m
0       1
1       1
2       2    1
3       6    3    1
4      24   12    4    6   1
5     120   60   20   30   5  10  1
6     720  360  120  180  30  60  6  90  15  20  1

Tilman Piesk, Triangle rows m=0..25, flattened
Tilman Piesk, Illustration of the multisets for m=0..6

Cf. A036038, A078760, A005651, A005651, A249620, A000041, A173333.

A047182 Number of nonempty subsets of {1,2,...,n} in which exactly 1/2 of the elements are <= (n-2)/2.

Original entry on oeis.org

0, 0, 0, 3, 4, 14, 20, 55, 83, 209, 329, 791, 1286, 3002, 5004, 11439, 19447, 43757, 75581, 167959, 293929, 646645, 1144065, 2496143, 4457399, 9657699, 17383859, 37442159, 67863914, 145422674, 265182524, 565722719, 1037158319

Offset: 1

Clark Kimberling

nonn

a(n) = C(n, [(n-2)/2]) - 1 = A037955(n) - 1. - Ralf Stephan, Mar 16 2004

A191528 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n that have k returns to the axis.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 1, 6, 3, 1, 10, 6, 3, 1, 20, 10, 4, 1, 35, 20, 10, 4, 1, 70, 35, 15, 5, 1, 126, 70, 35, 15, 5, 1, 252, 126, 56, 21, 6, 1, 462, 252, 126, 56, 21, 6, 1, 924, 462, 210, 84, 28, 7, 1, 1716, 924, 462, 210, 84, 28, 7, 1, 3432, 1716, 792, 330, 120, 36, 8, 1, 6435, 3432, 1716, 792, 330, 120, 36, 8, 1

Offset: 0

Emeric Deutsch, Jun 06 2011

Row n contains 1+floor(n/2) entries.
Sum of entries in row n is binomial(n, floor(n/2)) =A001405(n).
T(n,0) = A001405(n-1).
Rows 0, 2, 4, ... form triangle A100100.
Rows 1, 3, 5, ... form triangle A092392.
Sum_{k>=0} k*T(n,k) = A037955(n).
From Roger Ford, Oct 16 2020: (Start)
This is an empirical observation. T(n,k) = the number of different semi-meander arch depth models with n+2 top arches and k+1 arches at depth 0.  T(3,1) = the number of different semi-meander arch depth models with 5 top arches and 2 arches at depth 0.
Example: The depth of a semi-meander arch is the number of covering arches directly above the arch.  The arch depth model is the number of arches at each depth starting at 0 for a specific semi-meander.  The following is the arch depth models for semi-meanders with 5 top arches.
            /\                                   /\
           //\\                                 /  \
          ///\\\             depth             //\  \             depth
         ////\\\\  /\       (0)(1)(2)(3)      ///\\/\\ /\        (0)(1)(2)
depth    0123      0  model= 2  1  1  1       012  1   0   model= 2  2  1
         /\
        //\\    /\         depth         /\   /\            depth
       ///\\\  //\\       (0)(1)(2)     //\\ //\\ /\       (0)(1)
depth  012     01   model= 2  2  1      01   01   0  model= 3  2
         /\
        /  \               depth
       //\/\\ /\ /\       (0)(1)
depth  01 1   0  0  model= 3  2
There are 5 more semi-meanders with 5 top arches.  They are reflections of the above semi-meanders over a center vertical line and they yield the same arch depth models as the semi-meanders above.
T(3,1) = 2    different models=  2 2 1 and 2 1 1 1;
T(3,2) = 1     model=  3 2    (End).

			T(6,2)=3 because we have U(D)U(D)UU, U(D)UUD(D), and UUD(D)U(D), where U=(1,1) and D=(1,-1) (the return steps to the axis are shown between parentheses).
Triangle starts:
   1:
   1;
   1, 1;
   2, 1;
   3, 2, 1;
   6, 3, 1;
  10, 6, 3, 1;

Indranil Ghosh, Rows 0..100, flattened

Cf. A001405, A037955, A092392, A100100.

T := proc (n, k) if k <= floor((1/2)*n) then binomial(n-k-1, ceil((1/2)*n)-1) else 0 end if end proc: for n from 0 to 16 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form

Flatten[Table[Binomial[n-k-1,Ceiling[(n/2)-1]],{n,0,16},{k,0,Floor[n/2]}]] (* Indranil Ghosh, Mar 05 2017 *)

tabf(nn) = if(n==0, print1(1,", "), {for (n=1, nn, for(k=0, floor(n/2), print1(binomial(n-k-1, ceil((n/2)-1)),", ");); print();); });
tabf(16); \\ Indranil Ghosh, Mar 05 2017

T(n,k) = binomial(n-k-1, ceiling(n/2)-1) if 0 <= k <= floor(n/2).

G.f.: G(t,z) = 1/((1-z*c)*(1-t*z^2*c)), where c = (1-sqrt(1-4*z^2))/(2*z^2) is the Catalan function with argument z^2.

cp's OEIS Frontend

A000594 Ramanujan's tau function (or Ramanujan numbers, or tau numbers).

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A002054 Binomial coefficient C(2n+1, n-1).

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A061554 Square table read by antidiagonals: a(n,k) = binomial(n+k, floor(k/2)).

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A034869 Right half of Pascal's triangle.

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A101491 Triangle T(n,k), read by rows: number of Knödel walks starting at 0, ending at k, with n steps.

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