A035101 -id:A035101 - cp's OEIS frontend

A035342 The convolution matrix of the double factorial of odd numbers (A001147).

1, 3, 1, 15, 9, 1, 105, 87, 18, 1, 945, 975, 285, 30, 1, 10395, 12645, 4680, 705, 45, 1, 135135, 187425, 82845, 15960, 1470, 63, 1, 2027025, 3133935, 1595790, 370125, 43890, 2730, 84, 1, 34459425, 58437855, 33453945, 8998290

Offset: 1

Wolfdieter Lang

Previous name was: A triangle of numbers related to the triangle A035324; generalization of Stirling numbers of second kind A008277 and Lah numbers A008297.
If one replaces in the recurrence the '2' by '0', resp. '1', one obtains the Lah-number, resp. Stirling-number of 2nd kind, triangle A008297, resp. A008277.
The product of two lower triangular Jabotinsky matrices (see A039692 for the Knuth 1992 reference) is again such a Jabotinsky matrix: J(n,m) = Sum_{j=m..n} J1(n,j)*J2(j,m). The e.g.f.s of the first columns of these triangular matrices are composed in the reversed order: f(x)=f2(f1(x)). With f1(x)=-(log(1-2*x))/2 for J1(n,m)=|A039683(n,m)| and f2(x)=exp(x)-1 for J2(n,m)=A008277(n,m) one has therefore f2(f1(x))=1/sqrt(1-2*x) - 1 = f(x) for J(n,m)=a(n,m). This proves the matrix product given below. The m-th column of a Jabotinsky matrix J(n,m) has e.g.f. (f(x)^m)/m!, m>=1.
a(n,m) gives the number of forests with m rooted ordered trees with n non-root vertices labeled in an organic way. Organic labeling means that the vertex labels along the (unique) path from the root with label 0 to any leaf (non-root vertex of degree 1) is increasing. Proof: first for m=1 then for m>=2 using the recurrence relation for a(n,m) given below. - Wolfdieter Lang, Aug 07 2007
Also the Bell transform of A001147(n+1) (adding 1,0,0,... as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Matrix begins:
    1;
    3,   1;
   15,   9,   1;
  105,  87,  18,   1;
  945, 975, 285,  30,   1;
  ...
Combinatoric meaning of a(3,2)=9: The nine increasing path sequences for the three rooted ordered trees with leaves labeled with 1,2,3 and the root labels 0 are: {(0,3),[(0,1),(0,2)]}; {(0,3),[(0,2),(0,1)]}; {(0,3),(0,1,2)}; {(0,1),[(0,3),(0,2)]}; [(0,1),[(0,2),(0,3)]]; [(0,2),[(0,1),(0,3)]]; {(0,2),[(0,3),(0,1)]}; {(0,1),(0,2,3)}; {(0,2),(0,1,3)}.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Peter Bala, Generalized Dobinski formulas
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013-2014.
Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Richell O. Celeste, Roberto B. Corcino and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients. Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Tom Copeland, A Class of Differential Operators and the Stirling Numbers, 2015.
Tom Copeland, Addendum to Mathemagical Forests, 2010.
Tom Copeland, Mathemagical Forests, 2008.
Askar Dzhumadildaev and Damir Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) #09.8.3.
Donald E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA], 1992; Mathematica J. 2.1 (1992), no. 4, 67-78.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First 10 rows.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012.
Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. (2001) Vol. 239, 33-51.
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
Umesh Shankar, Log-concavity of rows of triangular arrays satisfying a certain super-recurrence, arXiv:2508.12467 [math.CO], 2025. See p. 4.

The column sequences are A001147, A035101, A035119, ...
Row sums: A049118(n), n >= 1.
Cf. A000108, A035324, A008277, A008297, A094638.

a035342 n k = a035342_tabl !! (n-1) !! (k-1)
a035342_row n = a035342_tabl !! (n-1)
a035342_tabl = map fst $ iterate (\(xs, i) -> (zipWith (+)
   ([0] ++ xs) $ zipWith (*) [i..] (xs ++ [0]), i + 2)) ([1], 3)
-- Reinhard Zumkeller, Mar 12 2014

T := (n,k) -> 2^(k-n)*hypergeom([k-n,k+1],[k-2*n+1],2)*GAMMA(2*n-k)/
(GAMMA(k)*GAMMA(n-k+1)); for n from 1 to 9 do seq(simplify(T(n,k)),k=1..n) od; # Peter Luschny, Mar 31 2015
T := (n, k) -> local j; 2^n*add((-1)^(k-j)*binomial(k, j)*pochhammer(j/2, n), j = 1..k)/k!: for n from 1 to 6 do seq(T(n, k), k=1..n) od;  # Peter Luschny, Mar 04 2024

a[n_, k_] := 2^(n+k)*n!/(4^n*n*k!)*Sum[(j+k)*2^(j)*Binomial[j + k - 1, k-1]*Binomial[2*n - j - k - 1, n-1], {j, 0, n-k}]; Flatten[Table[a[n,k], {n, 1, 9}, {k, 1, n}] ] [[1 ;; 40]] (* Jean-François Alcover, Jun 01 2011, after Vladimir Kruchinin *)

a(n,k):=2^(n+k)*n!/(4^n*n*k!)*sum((j+k)*2^(j)*binomial(j+k-1,k-1)*binomial(2*n-j-k-1,n-1),j,0,n-k); /* Vladimir Kruchinin, Mar 30 2011 */

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
print(bell_matrix(lambda n: A001147(n+1), 9)) # Peter Luschny, Jan 19 2016

a(n, m) = Sum_{j=m..n} |A039683(n, j)|*S2(j, m) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. to Wolfdieter Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the comment on products of Jabotinsky matrices.

a(n, m) = n!*A035324(n, m)/(m!*2^(n-m)), n >= m >= 1; a(n+1, m)= (2*n+m)*a(n, m)+a(n, m-1); a(n, m) := 0, n

E.g.f. of m-th column: ((x*c(x/2)/sqrt(1-2*x))^m)/m!, where c(x) = g.f. for Catalan numbers A000108.

From Vladimir Kruchinin, Mar 30 2011: (Start)

G.f. (1/sqrt(1-2*x) - 1)^k = Sum_{n>=k} (k!/n!)*a(n,k)*x^n.

a(n,k) = 2^(n+k) * n! / (4^n*n*k!) * Sum_{j=0..n-k} (j+k) * 2^(j) * binomial(j+k-1,k-1) * binomial(2*n-j-k-1,n-1). (End)

From Peter Bala, Nov 25 2011: (Start)

E.g.f.: G(x,t) = exp(t*A(x)) = 1 + t*x + (3*t + t^2)*x^2/2! + (15*t + 9*t^2 + t^3)*x^3/3! + ..., where A(x) = -1 + 1/sqrt(1-2*x) satisfies the autonomous differential equation A'(x) = (1+A(x))^3.

The generating function G(x,t) satisfies the partial differential equation t*(dG/dt+G) = (1-2*x)*dG/dx, from which follows the recurrence given above.

The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^3*d/dx. Cf. A008277 (D = (1+x)*d/dx), A105278 (D = (1+x)^2*d/dx), A035469 (D = (1+x)^4*d/dx) and A049029 (D = (1+x)^5*d/dx). (End)

The n-th row polynomial R(n,x) is given by the Dobinski-type formula R(n,x) = exp(-x)*Sum_{k>=1} k*(k+2)*...*(k+2*n-2)*x^k/k!. - Peter Bala, Jun 22 2014

T(n,k) = 2^(k-n)*hypergeom([k-n,k+1],[k-2*n+1],2)*Gamma(2*n-k)/(Gamma(k)*Gamma(n-k+1)). - Peter Luschny, Mar 31 2015

T(n,k) = 2^n*Sum_{j=1..k} ((-1)^(k-j)*binomial(k, j)*Pochhammer(j/2, n)) / k!. - Peter Luschny, Mar 04 2024

Simpler name from Peter Luschny, Mar 31 2015

A180048 Coefficient triangle of the denominators of the (n-th convergents to) the continued fraction 1/(w+2/(w+3/(w+4/... . Conjectured to equal unsigned version of A137286.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 5, 0, 1, 8, 0, 9, 0, 1, 0, 33, 0, 14, 0, 1, 48, 0, 87, 0, 20, 0, 1, 0, 279, 0, 185, 0, 27, 0, 1, 384, 0, 975, 0, 345, 0, 35, 0, 1, 0, 2895, 0, 2640, 0, 588, 0, 44, 0, 1, 3840, 0, 12645, 0, 6090, 0, 938, 0, 54, 0, 1, 0, 35685, 0, 41685, 0, 12558, 0, 1422, 0, 65

Offset: 0

Wouter Meeussen, Aug 08 2010

Equivalence to the recurrence formula needs formal proof. This continued fraction converges to 0.525135276160981... for w=1. A conjecture by Ramanujan puts this equal to -1 + 1/(sqrt(e*Pi/2) - Sum_{k>=1} 1/(2k-1)!!).
From Alexander Kreinin, Oct 26 2015: (Start)
Let us denote the continued fraction by U(w).
Then it is easy to show that Mill's ratio, R(w) = (1 - Phi(w))/f(w), where Phi is the standard normal distribution function and f is the standard normal density function, satisfies R(w) = 1/(w + U(w)).
Indeed, R(w) = 1/(w+1/(w+2/(w+3/(w+... Then we find U(w) = 1/R(w) - w. It was proved in Alexander Kreinin (arXiv:1405.5852) that R(w+t) + Q(w, t) = exp(w*t + w^2/2)*R(t), where Q(w,t) = Sum_{k>=0} Sum_{m=0..k} q(k,m) * t^m * w^(k+1)/(k+1)!.
Substituting t=0, we obtain R(w) = exp(w^2/2)*sqrt(Pi/2) - Sum_{n>=0} w^(2n+1)/(2n+1)!!. If w=1 we obtain Ramanujan's formula. (End)

			The denominator of 1/(w+2/(w+3/(w+4/(w+5/(w+6/w))))) equals 48 + 87w^2 + 20w^4 + w^6.
From _Joerg Arndt_, Apr 20 2013: (Start)
Triangle begins
     1;
     0,     1;
     2,     0,     1;
     0,     5,     0,     1;
     8,     0,     9,     0,    1;
     0,    33,     0,    14,    0,   1;
    48,     0,    87,     0,   20,   0,   1;
     0,   279,     0,   185,    0,  27,   0,  1;
   384,     0,   975,     0,  345,   0,  35,  0,  1;
     0,  2895,     0,  2640,    0, 588,   0, 44,  0, 1;
  3840,     0, 12645,     0, 6090,   0, 938,  0, 54, 0, 1;
     0, 35685,     0, 41685,    0, ... (End)

G. C. Greubel, Table of n, a(n) for the first 75 rows, flattened
Authors?, Hungarian discussion forum
Alexander Kreinin, Combinatorial Properties of Mills' Ratio, arXiv:1405.5852 [math.CO], 2014. See Table 3.
Alexander Kreinin, Integer Sequences and Laplace Continued Fraction, preprint, 2016.
Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences 19 (2016), Article 16.6.2.

Cf. A137286, A084950, A180047, A180049.

Cf. A111129.

Table[ CoefficientList[ Denominator[ Together[ Fold[ #2/(w+#1) &, Infinity, Reverse @ Table[ k, {k, 1, n} ] ] ] ], w ], {n, 16} ] (* or equivalently *) Clear[ p ];p[ 0 ]=1; p[ 1 ]=w; p[ n_ ]:=p[ n ]= w*p[ n-1 ] + n*p[ n-2 ]; Table[ CoefficientList[ p[ k ]//Expand, w ], {k,0,15} ]

p(0)=1; p(1)=w; p(n) = w*p(n-1) + n*p(n-2) (conjecture).
T(n,k) = T(n-1,k-1) + n*T(n-2,k), T(0,0) = 1, T(1,0) = 0, T(1,1) = 1. - Philippe Deléham, Oct 28 2013
sum_{k=0..n} T(n,k) = A000932(n). - Philippe Deléham, Oct 28 2013
T(2n,0) = A000165(n); T(2n+1,1) = A129890(n); T(2n+2,2) = A035101(n+2). - Philippe Deléham, Oct 28 2013

A336599 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords are contained within the marked chord.

Original entry on oeis.org

1, 5, 1, 33, 9, 3, 279, 87, 39, 15, 2895, 975, 495, 255, 105, 35685, 12645, 6885, 4005, 2205, 945, 509985, 187425, 106785, 66465, 41265, 23625, 10395, 8294895, 3133935, 1843695, 1198575, 795375, 513135, 301455, 135135, 151335135, 58437855, 35213535, 23601375, 16343775, 11263455, 7453215, 4459455, 2027025

Offset: 1

Donovan Young, Jul 29 2020

			Triangle begins:
     1;
     5,    1;
    33,    9,    3;
   279,   87,   39,  15;
  2895,  975,  495, 255, 105;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can only be (1,4) and it contains one other chord, namely (2,3), hence T(2,1) = 1.

Donovan Young, Table of n, a(n) for n = 1..9870
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Row sums are n*A001147(n) for n > 0.
Leading diagonal is A001147(n-1) for n > 0.
The first column is A129890(n-1) for n > 0.
The second column is A035101(n+1) for n > 0.
Cf. A336598, A336600, A336601.

CoefficientList[Normal[Series[(Sqrt[1-2*y*x]-Sqrt[1-2*x])/(1-2*x)/(1-y),{x,0,10}]]/.{x^n_.->x^n*n!},{x,y}]

E.g.f.: (sqrt(1 - 2*y*x) - sqrt(1 - 2*x))/(1 - 2*x)/(1 - y).

A263384 Fourth column of the matrix of polynomial coefficients of the rational approximation to Mill's ratio.

Original entry on oeis.org

1, 14, 185, 2640, 41685, 729330, 14073885, 297693900, 6859400625, 171172905750, 4601737965825, 132643472761800, 4082080279402125, 133614981594344250, 4635763624512145125, 169957871025837394500

Offset: 0

Alexander Kreinin, Oct 16 2015

Rational approximations, Q_{k-1}(t)/P_k(t), to Mill's ratio, R(t)=(1-Phi(t))/f(t), where Phi(t) is the standard normal distribution function and f(t) is the standard normal density, were discovered by Laplace, who computed the first four polynomials. Thirty years later, Jacobi derived recurrence relations for these polynomials and analyzed some of their analytical properties. The coefficients q_{k,m} of Q_k(t) form a matrix, of which this is the fourth column. The double generating function for the polynomials Q_k(t) is computed in A. Kreinin (see Links). The coefficients q_{k,m} are described by the triangular array A180048.

Selden Crary, Richard Diehl Martinez, Michael Saunders, The Nu Class of Low-Degree-Truncated Rational Multifunctions. Ib. Integrals of Matern-correlation functions for all odd-half-integer class parameters, arXiv:1707.00705 [stat.ME], 2017, Table 2.
A. Kreinin, Combinatorial properties of the Mills Ratio, arXiv:1405.5852 [math.CO], 2014.
Alexander Kreinin, Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity, Journal of Integer Sequences, 19 (2016), #16.6.2.

Columns of the matrix [q_{k,m}] include: A000165 (m=1), A129890 (m=2), A035101 (m=3), this sequence (m=4).

Cf. A180048.

Table[((2 n + 6)!! - 3 (2 n + 5)!! + (2 n + 3)!!)/6, {n, 0, 12}] (* Michael De Vlieger, Oct 27 2015 *)

a(n)=(prod(k=1, n+3, 2*k)-3*prod(k=1, n+3,(2*k-1))+prod(k=1, n+2, 2*k-1))/6;
vector(20, n, a(n-1)) \\ Altug Alkan, Oct 16 2015

a(n) = ((2*n+6)!! - 3*(2*n+5)!! + (2*n+3)!!)/6, n>=0.

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A035119 Related to A045720 and A035101.

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A035342 The convolution matrix of the double factorial of odd numbers (A001147).

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A180048 Coefficient triangle of the denominators of the (n-th convergents to) the continued fraction 1/(w+2/(w+3/(w+4/... . Conjectured to equal unsigned version of A137286.

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A336599 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords are contained within the marked chord.

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A263384 Fourth column of the matrix of polynomial coefficients of the rational approximation to Mill's ratio.

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