A132062 Sheffer triangle (1,1-sqrt(1-2*x)). Extended Bessel triangle A001497.
1, 0, 1, 0, 1, 1, 0, 3, 3, 1, 0, 15, 15, 6, 1, 0, 105, 105, 45, 10, 1, 0, 945, 945, 420, 105, 15, 1, 0, 10395, 10395, 4725, 1260, 210, 21, 1, 0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1, 0, 2027025, 2027025, 945945, 270270, 51975, 6930, 630, 36, 1, 0
Offset: 0
Examples
[1] [0, 1] [0, 1, 1] [0, 3, 3, 1] [0, 15, 15, 6, 1] [0, 105, 105, 45, 10, 1] [0, 945, 945, 420, 105, 15, 1] [0, 10395, 10395, 4725, 1260, 210, 21, 1] [0, 135135, 135135, 62370, 17325, 3150, 378, 28, 1]
References
- Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From N. J. A. Sloane, Sep 15 2012
- Steven Roman, The Umbral Calculus, Pure and Applied Mathematics, 111, Academic Press, 1984. (p. 78) [Emanuele Munarini, Oct 10 2017]
Links
- Leonard Carlitz, A Note on the Bessel Polynomials, Duke Math. J. 24 (2) (1957), 151-162. [_Emanuele Munarini_, Oct 10 2017]
- H. Han and S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From _R. J. Mathar_, Mar 20 2009]
- Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
- Wolfdieter Lang, First 10 rows.
- Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 18.
Crossrefs
Programs
-
Maple
# The function BellMatrix is defined in A264428. BellMatrix(n -> doublefactorial(2*n-1), 9); # Peter Luschny, Jan 27 2016 # Alternative: egf := exp(y*(1 - sqrt(1 - 2*x))): serx := series(egf, x, 12): coefx := n -> n!*coeff(serx, x, n): row := n -> seq(coeff(coefx(n), y, k), k = 0..n): for n from 0 to 8 do row(n) od; # Peter Luschny, Apr 25 2024
-
Mathematica
Table[If[k <= n, Binomial[2n-2k,n-k] Binomial[2n-k-1,k-1] (n-k)!/2^(n-k), 0], {n, 0, 6}, {k, 0, n}] // Flatten (* Emanuele Munarini, Oct 10 2017 *) BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]]; rows = 10; M = BellMatrix[(2#-1)!!&, rows]; Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *) -
Sage
# uses[bell_transform from A264428] def A132062_row(n): a = sloane.A001147 dblfact = a.list(n) return bell_transform(n, dblfact) [A132062_row(n) for n in (0..9)] # Peter Luschny, Dec 20 2015
Formula
a(n,m)=0 if n
E.g.f. m-th column ((x*f2p(1;x))^m)/m!, m>=0. with f2p(1;x):=1-sqrt(1-2*x)= x*c(x/2) with the o.g.f.of A000108 (Catalan).
From Emanuele Munarini, Oct 10 2017: (Start)
a(n,k) = binomial(2*n-2*k,n-k)*binomial(2*n-k-1,k-1)*(n-k)!/2^(n-k).
The row polynomials p_n(x) (studied by Carlitz) satisfy the recurrence: p_{n+2}(x) - (2*n+1)*p_{n+1}(x) - x^2*p_n(x) = 0. (End)
T(n, k) = n! [y^k] [x^n] exp(y*(1 - sqrt(1 - 2*x))). - Peter Luschny, Apr 25 2024
A293157 Triangle read by rows: T(n,k) = number of linear chord diagrams with n chords such that every chord has length at least k (1 <= k <= n).
1, 3, 1, 15, 5, 1, 105, 36, 10, 1, 945, 329, 99, 20, 1, 10395, 3655, 1146, 292, 40, 1, 135135, 47844, 15422, 4317, 876, 80, 1, 2027025, 721315, 237135, 69862, 16924, 2628, 160, 1, 34459425, 12310199, 4106680, 1251584, 332507, 67404, 7884, 320, 1, 654729075, 234615096, 79154927, 24728326, 6944594, 1627252, 269616, 23652, 640, 1
Offset: 1
Comments
There is a surprising change in notation in Sullivan (2016) between Definition 1 and Table 1.
The first 11 columns are given in the reference.
Examples
Triangle begins:
1;
3, 1;
15, 5, 1;
105, 36, 10, 1;
945, 329, 99, 20, 1;
10395, 3655, 1146, 292, 40, 1;
...
Links
- Everett Sullivan, Linear chord diagrams with long chords, arXiv preprint arXiv:1611.02771 [math.CO], 2016.
Extensions
More terms from Alois P. Heinz, Oct 17 2017
A190823 Number of permutations of 2 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 2.
1, 0, 0, 1, 10, 99, 1146, 15422, 237135, 4106680, 79154927, 1681383864, 39034539488, 983466451011, 26728184505750, 779476074425297, 24281301468714902, 804688068731837874, 28269541494090294129, 1049450257149017422000, 41050171013933837206545
Offset: 0
Comments
From Gus Wiseman, Feb 27 2019: (Start)
Also the number of 2-uniform set partitions of {1..2n} such that no block has its two vertices differing by less than 3. For example, the a(4) = 10 set partitions are:
{{1,4}, {2,6}, {3,7}, {5,8}}
{{1,4}, {2,7}, {3,6}, {5,8}}
{{1,5}, {2,6}, {3,7}, {4,8}}
{{1,5}, {2,6}, {3,8}, {4,7}}
{{1,5}, {2,7}, {3,6}, {4,8}}
{{1,5}, {2,8}, {3,6}, {4,7}}
{{1,6}, {2,5}, {3,7}, {4,8}}
{{1,6}, {2,5}, {3,8}, {4,7}}
{{1,7}, {2,5}, {3,6}, {4,8}}
{{1,8}, {2,5}, {3,6}, {4,7}}
(End)
Examples
All solutions for n=4 (read downwards): 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 1 4 4 1 4 4 1 1 2 1 4 2 1 4 2 2 3 3 1 2 2 3 2 3 1 3 2 4 4 4 3 4 3 2 3 1 4 2 3 3 4 1 4 4 4 4
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..404
- Dmitry Efimov, Hafnian of two-parameter matrices, arXiv:2101.09722 [math.CO], 2021.
- Everett Sullivan, Linear chord diagrams with long chords, arXiv preprint arXiv:1611.02771 [math.CO], 2016.
Crossrefs
Programs
-
Magma
I:=[1,0,0,1,10,99]; [n le 5 select I[n] else 2*n*Self(n-1) -2*(3*n-8)*Self(n-2) +2*(3*n-11)*Self(n-3) -2*(n-5)*Self(n-4) -Self(n-5): n in [1..40]]; // G. C. Greubel, Dec 03 2023
-
Mathematica
a[0]=1; a[1]=0; a[2]=0; a[3]=1; a[4]=10; a[5]=99; a[n_] := a[n] = (2*n+2) a[n-1] - (6*n-10) a[n-2] + (6*n-16) a[n-3] - (2*n-8) a[n-4] - a[n-5]; Array[a, 20, 0] (* based on Sullivan's formula, Giovanni Resta, Mar 20 2017 *) dtui[{}]:={{}};dtui[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@dtui[Complement[set,s]]]/@Table[{i,j},{j,Select[set,#>i+2&]}]; Table[Length[dtui[Range[n]]],{n,0,12,2}] (* Gus Wiseman, Feb 27 2019 *)
-
SageMath
@CachedFunction def a(n): # a = A190823 if (n<6): return (1,0,0,1,10,99)[n] else: return 2*(n+1)*a(n-1) - 2*(3*n-5)*a(n-2) + 2*(3*n-8)*a(n-3) - 2*(n-4)*a(n-4) - a(n-5) [a(n) for n in range(41)] # G. C. Greubel, Dec 03 2023
Formula
a(n) = 2*(n+1)*a(n-1) - 2*(3*n-5)*a(n-2) + 2*(3*n-8)*a(n-3) - 2*(n-4)*a(n-4) - a(n-5) (proved). - Everett Sullivan, Mar 16 2017
a(n) ~ 2^(n+1/2) * n^n / exp(n+2), based on Sullivan's formula. - Vaclav Kotesovec, Mar 21 2017
Extensions
a(16)-a(20) (using Everett Sullivan's formula) from Giovanni Resta, Mar 20 2017
a(0)=1 prepended by Alois P. Heinz, Oct 17 2017
A265192 T(n,k)=Number of nXk arrays containing k copies of 0..n-1 with no equal horizontal or antidiagonal neighbors and new values introduced sequentially from 0.
1, 0, 1, 0, 1, 1, 0, 0, 5, 1, 0, 1, 17, 36, 1, 0, 0, 42, 774, 329, 1, 0, 1, 155, 20592, 76035, 3655, 1, 0, 0, 511, 583806, 20957398, 10866362, 47844, 1, 0, 1, 2023, 17355854, 6394366422, 38833756515, 2130866037, 721315, 1, 0, 0, 7760, 531710144
Offset: 1
Comments
Table starts
.1........0............0...............0...............0..................0
.1........1............0...............1...............0..................1
.1........5...........17..............42.............155................511
.1.......36..........774...........20592..........583806...........17355854
.1......329........76035........20957398......6394366422......2057050979371
.1.....3655.....10866362.....38833756515.157027988934320.683255971286971494
.1....47844...2130866037.117650546564305
.1...721315.551607137250
.1.12310199
Examples
Some solutions for n=4 k=4 ..0..1..2..3....0..1..0..2....0..1..2..3....0..1..2..3....0..1..2..1 ..2..3..0..2....3..2..1..3....0..3..1..0....3..0..2..1....2..0..3..0 ..1..3..0..1....0..3..2..1....2..3..1..2....2..0..3..2....3..1..2..3 ..1..3..0..2....0..1..3..2....2..3..1..0....1..0..3..1....2..0..1..3
Links
- R. H. Hardin, Table of n, a(n) for n = 1..61
Crossrefs
Column 2 is A000806.
A232690 E.g.f. satisfies: A(x) = exp( 1/A(x) * Integral A(x)^3 dx ).
1, 1, 2, 7, 33, 202, 1495, 13107, 132062, 1508629, 19227687, 270818542, 4173948097, 69906444393, 1263811926338, 24534217063999, 508951297964193, 11236656534791578, 263054502440239639, 6508910392250017611, 169727899004807970782, 4652123984505282141277, 133711980572082349859559
Offset: 0
Keywords
Comments
Note that G(x) = exp(1/G(x) * Integral G(x)^2 dx) has negative coefficients.
Compare e.g.f. to: B(x) = exp( 1/B(x) * Integral B(x) dx ) where B(y) = Bessel polynomial y_n(-1) (cf. A000806).
Examples
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 33*x^4/4! + 202*x^5/5! +... Related expansions: log(A(x)) = x + x^2/2! + 3*x^3/3! + 11*x^4/4! + 61*x^5/5! + 393*x^6/6! +... Integral A(x)^3 dx = x + 3*x^2/2! + 12*x^3/3! + 63*x^4/4! + 411*x^5/5! +... 1/A(x) = 1 - x - x^3/3! - x^4/4! - 12*x^5/5! - 41*x^6/6! - 451*x^7/7! -...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..100
Programs
-
Maple
seq(n! * coeff(series(sqrt(LambertW(-1,(4*x-3)*exp(-3))/(4*x-3)), x, n+1), x, n), n=0..20); # Vaclav Kotesovec, Jan 05 2014
-
Mathematica
CoefficientList[FullSimplify[Assuming[Element[x, Reals], Series[Sqrt[LambertW[-1,(4*x-3)*E^(-3)]/(4*x-3)], {x, 0, 20}]]], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 05 2014 *) -
PARI
{a(n)=local(A=1+x);for(i=1,n,A=exp(1/A*intformal(A^3+x*O(x^n))));n!*polcoeff(A,n)} for(n=0,30,print1(a(n),", "))
Formula
E.g.f.: sqrt(LambertW(-1,(4*x-3)*exp(-3))/(4*x-3)). - Vaclav Kotesovec, Jan 05 2014
Limit n->infinity (a(n)/n!)^(1/n) = 4/3. - Vaclav Kotesovec, Jan 05 2014
A006146 Sums of prime divisors of Ruth-Aaron numbers (A006145).
5, 5, 7, 18, 15, 20, 44, 46, 29, 31, 50, 30, 20, 34, 75, 162, 146, 46, 14, 113, 53, 66, 333, 36, 514, 318, 43, 193, 279, 418, 30, 121, 55, 485, 200, 136, 77, 37, 211, 587, 147, 269, 477, 108, 136, 235, 185, 290, 333, 309, 493, 177, 199, 223, 641, 531, 182, 368
Offset: 1
Keywords
References
- John L. Drost, Ruth/Aaron Pairs, J. Recreational Math. 28 (No. 2), 120-122.
- Dana Mackenzie, Homage to an itinerant master, Science, vol. 275, p. 759, 1997.
- Carol Nelson, David E. Penney, and Carl Pomerance, 714 and 715. Journal of Recreational Mathematics 7(2):87-89, 1974.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74. (Annotated scanned copy)
- Ivars Peterson, Playing with Ruth-Aaron pairs
- Ivars Peterson's MathTrek, Playing with Ruth-Aaron Pairs [In the internet archive]
- Eric Weisstein's World of Mathematics, Ruth-Aaron Pair
Programs
-
Maple
with(numtheory): for n from 1 to 10000 do t0 := 0; t1 := factorset(n); for j from 1 to nops(t1) do t0 := t0+t1[ j ]; od: s[ n ] := t0; od: for n from 1 to 9999 do if s[ n ] = s[ n+1 ] then lprint(n,s[ n ]); fi; od:
-
Mathematica
Cases[Partition[(Plus@@(First@#&/@FactorInteger@#)&/@Range@100000),2,1],{a_,a_}:>a] (* Hans Rudolf Widmer, May 31 2024 *) -
Python
from sympy import primefactors def aupton(terms): alst, k, sopfk, sopfkp1 = [], 0, 0, 1 while len(alst) < terms: k, sopfk, sopfkp1 = k+1, sopfkp1, sum(p for p in primefactors(k+1)) if sopfkp1 == sopfk: alst.append(sopfk) return alst print(aupton(58)) # Michael S. Branicky, May 05 2021
Formula
A006199 Bessel polynomial {y_n}'(-1).
0, 1, -3, 21, -185, 2010, -25914, 386407, -6539679, 123823305, -2593076255, 59505341676, -1484818160748, 40025880386401, -1159156815431055, 35891098374564105, -1183172853341759129, 41372997479943753582, -1529550505546305534414, 59608871544962952539335
Offset: 1
Keywords
Comments
Absolute values give partitions into pairs.
References
- G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74.
- J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
Programs
-
Mathematica
Join[{0}, Table[2*n*Pochhammer[1/2, n]*(-2)^(n - 1)* Hypergeometric1F1[1 - n, -2*n, -2], {n,1,50}]] (* G. C. Greubel, Aug 14 2017 *) -
PARI
for(n=0,50, print1(sum(k=0,n-1, ((n+k)!/(k!*(n-k)!))*(-1/2)^k), ", ")) \\ G. C. Greubel, Aug 14 2017
Formula
From G. C. Greubel, Aug 14 2017: (Start)
a(n) = 2*n*(1/2){n} * (-2)^(n-1) * hyergeometric1f1(1-n; -2*n; -2), where (a){n} is the Pochhammer symbol.
E.g.f.: (1+2*x)^(-3/2)*( (1+2*x)^(3/2) - x*(1+2*x)^(1/2) - x -1) * exp(sqrt(1+2*x) - 1), for offset 0. (End)
G.f.: (x/(1-x)^3)*hypergeometric2f0(2,3/2; - ; -2*x/(1-x)^2), for offset 0. - G. C. Greubel, Aug 16 2017
A101682 Expansion of 2 - exp(-1 + sqrt(1-4x)).
1, 2, 0, 8, 80, 1152, 21056, 467840, 12248064, 369313280, 12605643776, 480491716608, 20231074672640, 932551401807872, 46708494389084160, 2525988902617776128, 146694190329387352064, 9105143756032486932480
Offset: 0
Keywords
Crossrefs
Cf. A000806.
A122850 Exponential Riordan array (1, sqrt(1+2x)-1).
1, 0, 1, 0, -1, 1, 0, 3, -3, 1, 0, -15, 15, -6, 1, 0, 105, -105, 45, -10, 1, 0, -945, 945, -420, 105, -15, 1, 0, 10395, -10395, 4725, -1260, 210, -21, 1, 0, -135135, 135135, -62370, 17325, -3150, 378, -28, 1, 0, 2027025, -2027025, 945945, -270270, 51975, -6930, 630, -36, 1
Offset: 0
Comments
Inverse of number triangle A122848. Entries are Bessel polynomial coefficients. Row sums are A000806.
Also the inverse Bell transform of the sequence "g(n) = 1 if n<2 else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016
Examples
Triangle begins 1 0 1 0 -1 1 0 3 -3 1 0 -15 15 -6 1 0 105 -105 45 -10 1 0 -945 945 -420 105 -15 1 0 10395 -10395 4725 -1260 210 -21 1 0 -135135 135135 -62370 17325 -3150 378 -28 1 0 2027025 -2027025 945945 -270270 51975 -6930 630 -36 1 0 -34459425 34459425 -16216200 4729725 -945945 135135 -13860 990 -45 1 ...
Links
- P. Bala, The white diamond product of power series
- Orli Herscovici, Study of the p,q-deformed Touchard polynomials, arXiv:1904.07674 [math.CO], 2019.
- M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
- Wikipedia, Bessel polynomials
- S. Willerton, The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials, arXiv:1708.03227v1 [math.MG], 2017.
Programs
-
Maple
# The function BellMatrix is defined in A264428. BellMatrix(n -> (-1)^n*doublefactorial(2*n-1), 9); # Peter Luschny, Jan 27 2016
-
Mathematica
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]]; rows = 12; M = BellMatrix[Function[n, (-1)^n (2n-1)!!], rows]; Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 26 2018, after Peter Luschny *) -
Sage
# uses[bell_matrix from A264428] bell_matrix(lambda n: 1 if n<2 else 0, 12).inverse() # Peter Luschny, Jan 19 2016
Formula
T(n,k) = (-1)^(n-k)*A132062(n,k). - Philippe Deléham, Nov 06 2011
Triangle equals the matrix product A039757*A008277. Dobinski-type formula for the row polynomials: R(n,x) = x*exp(-x)*Sum_{k = 0..inf} (k-1)*(k-3)*(k-5)*...*(k-(2*n-3))*x^k/k! for n >= 1. Cf. A001497. - Peter Bala, Jun 23 2014
From Peter Bala, Jan 09 2018: (Start)
Alternative Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k = 0..inf} k*(k-2)*(k-4)*...*(k-(2*n-2))*x^k/k!.
Equivalently, R(n,x) = x o (x-2) o (x-4) o...o (x-(2*n-2)), where o denotes the white diamond product of polynomials. See the Bala link for the definition and details.
The white diamond products (x-1) o (x-3) o...o (x-(2*n-3)) give the row polynomials of the array with a factor of x removed.
If d is the first derivative operator f -> d/dx(f(x)) and D is the operator f(x) -> 1/x*d/dx(f(x)) then x^(2*n)*D^n = R(n,x*d), with the understanding that (x*d)^k is to interpreted as the operator f(x) -> x^k*d^k(f(x))/dx^k. (End)
Sum_{k=0..n} (-1)^(n+k) * T(n,k) = A144301(n). - Alois P. Heinz, Aug 31 2022
Extensions
More terms from Alois P. Heinz, Aug 31 2022
A380208 Expansion of e.g.f. exp( (1+3*x)^(1/3) - 1 ).
1, 1, -1, 5, -39, 421, -5809, 97609, -1933455, 44107881, -1138752449, 32820576141, -1044523471991, 36379398867085, -1376300966184689, 56200996031812241, -2463713702730471199, 115400572452587463249, -5751849729149085927425, 303954806150664749166101
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..381
Programs
-
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1+3*x)^(1/3)-1)))
Formula
a(n) = Sum_{k=0..n} 3^(n-k) * Stirling1(n,k) * Bell(k).
a(n) = (1/e) * 3^n * n! * Sum_{k>=0} binomial(k/3,n)/k!.
a(0) = 1; a(n) = Sum_{k=1..n} (Product_{j=0..k-1} (-3*j+1)) * binomial(n-1,k-1) * a(n-k).
Comments