A030037
a(n+1) = Sum_{k=0..floor(2*n/5)} a(k) * a(n-k).
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 6, 11, 20, 40, 77, 160, 319, 636, 1332, 2721, 5799, 12068, 25109, 53943, 113682, 245931, 523896, 1115239, 2425858, 5208339, 11388934, 24630843, 53194684, 116764483, 253764437, 559289434, 1221970242, 2666776056, 5889628123, 12910470041, 28608337855
Offset: 0
-
lista(nn) = {my(v=vector(nn+2, i, 1)); for(n=2, nn, v[n+2]=sum(k=1, 1+(2*n)\5, v[k]*v[n+2-k])); v; } \\ Jinyuan Wang, Mar 18 2020
A030038
a(n+1) = Sum_{k=0..floor(3*n/5)} a(k) * a(n-k).
Original entry on oeis.org
1, 1, 1, 2, 3, 6, 13, 26, 57, 117, 262, 602, 1329, 3079, 6903, 16161, 38185, 87745, 207896, 481817, 1148246, 2755781, 6485489, 15592656, 36899080, 88966356, 215687474, 515590781, 1251629165, 2998835401, 7291353036, 17803231663, 42980538045, 104981503048, 253813177447
Offset: 0
-
lista(nn) = {my(v=vector(nn+2, i, 1)); for(n=1, nn, v[n+2]=sum(k=1, 1+(3*n)\5, v[k]*v[n+2-k])); v; } \\ Jinyuan Wang, Mar 18 2020
A300440
Number of odd strict trees of weight n (all outdegrees are odd).
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 2, 3, 5, 7, 11, 18, 27, 45, 75, 125, 207, 353, 591, 1013, 1731, 2984, 5122, 8905, 15369, 26839, 46732, 81850, 142932, 251693, 441062, 778730, 1370591, 2425823, 4281620, 7601359, 13447298, 23919512, 42444497, 75632126, 134454505, 240100289
Offset: 1
The a(10) = 7 odd strict trees: 10, (721), (631), (541), (532), ((421)21), ((321)31).
Cf.
A000009,
A000992,
A032305,
A063834,
A078408,
A089259,
A196545,
A273873,
A279785,
A289501,
A298118,
A300301,
A300352,
A300353,
A300436,
A300439.
-
g[n_]:=g[n]=1+Sum[Times@@g/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&&UnsameQ@@#&]}];
Array[g,20]
-
seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)) - prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)/2); v} \\ Andrew Howroyd, Aug 25 2018
A030032
a(n+1) = Sum_{k=0..floor(n/3)} a(k) * a(n-k).
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 5, 10, 18, 33, 66, 127, 244, 506, 1009, 2018, 4229, 8631, 17631, 37265, 77349, 160433, 342807, 720334, 1513791, 3254445, 6902283, 14634519, 31602375, 67522598, 144182089, 312851626, 672411931, 1444776938, 3145883976, 6794243911, 14667444523
Offset: 0
-
lista(nn) = {my(v=vector(nn+2, i, 1)); for(n=3, nn, v[n+2]=sum(k=1, 1+n\3, v[k]*v[n+2-k])); v; } \\ Jinyuan Wang, Mar 18 2020
A030034
a(n+1) = Sum_{k=0..floor(n/4)} a(k) * a(n-k).
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 3, 5, 8, 16, 29, 53, 98, 196, 376, 723, 1393, 2786, 5474, 10752, 21128, 42979, 85905, 171810, 343522, 703270, 1421998, 2875847, 5817194, 11978061, 24428369, 49817625, 101588512, 209911975, 430722621, 883632025, 1812556425, 3763773324, 7764267706
Offset: 0
-
lista(nn) = {my(v=vector(nn+2, i, 1)); for(n=3, nn, v[n+2]=sum(k=1, 1+n\4, v[k]*v[n+2-k])); v; } \\ Jinyuan Wang, Mar 18 2020
A030039
a(n+1) = Sum_{k=0..floor(4*n/5)} a(k) * a(n-k).
Original entry on oeis.org
1, 1, 1, 2, 4, 9, 21, 42, 97, 233, 570, 1414, 3319, 8208, 20595, 52370, 133955, 328261, 833652, 2136155, 5519116, 14341917, 36069212, 93074465, 241613666, 630653714, 1652472131, 4215072482, 10985970447, 28756470626, 75580683613, 199201407564, 513021407584
Offset: 0
-
lista(nn) = {my(v=vector(nn+2, i, 1)); for(n=1, nn, v[n+2]=sum(k=1, 1+(4*n)\5, v[k]*v[n+2-k])); v; } \\ Jinyuan Wang, Mar 18 2020
A124973
a(n) = Sum_{k=0..(n-2)/2} a(k)a*(n-1-k), with a(0) = a(1) = 1.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 6, 11, 22, 42, 87, 174, 365, 745, 1587, 3303, 7103, 14974, 32477, 69284, 151172, 325077, 713400, 1545719, 3406989, 7423648, 16429555, 35992438, 79912474, 175785514, 391488688, 864591621, 1930333822, 4276537000
Offset: 0
-
a:= proc(n) option remember;
if n<2 then 1
else add(a(j)*a(n-j-1), j=0..floor((n-2)/2))
fi
end:
seq(a(n), n=0..40); # G. C. Greubel, Nov 19 2019
-
a[n_]:= a[n]= If[n<2, 1, Sum[a[j]*a[n-j-1], {j, 0, (n-2)/2}]]; Table[a[n], {n, 0, 40}] (* G. C. Greubel, Nov 19 2019 *)
-
a(n) = if(n<2, 1, sum(j=0, (n-2)\2, a(j)*a(n-j-1))); \\ G. C. Greubel, Nov 19 2019
-
@CachedFunction
def a(n):
if (n<2): return 1
else: return sum(a(j)*a(n-j-1) for j in (0..floor((n-2)/2)))
[a(n) for n in (0..40)] # G. C. Greubel, Nov 19 2019
A300864
Signed recurrence over strict trees: a(n) = -1 + Sum_{y1 + ... + yk = n, y1 > ... > yk > 0, k > 1} a(y1) * ... * a(yk).
Original entry on oeis.org
1, 1, 0, 1, 0, 0, 2, -1, 0, 4, -6, 6, 6, -24, 38, -17, -64, 188, -230, -6, 662, -1432, 1286, 1210, -6362, 10692, -5530, -18274, 57022, -74364, 174, 216703, -489544, 467860, 391258, -2256430, 3948206, -2234064, -6725362, 21920402, -29716570, 2095564, 84595798, -198418242, 197499846
Offset: 1
Cf.
A000992,
A018819,
A063834,
A099323,
A196545,
A220418,
A273866,
A273873,
A289501,
A290261,
A300862,
A300863,
A300865,
A300866.
-
a[n_]:=a[n]=-1+Sum[Times@@a/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&UnsameQ@@#&]}];
-Array[a,40]
A300865
Signed recurrence over binary enriched p-trees: a(n) = (-1)^(n-1) + Sum_{x + y = n, 0 < x <= y < n} a(x) * a(y).
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 2, 2, 4, 6, 10, 16, 27, 46, 77, 131, 224, 391, 672, 1180, 2050, 3626, 6344, 11276, 19863, 35479, 62828, 112685, 200462, 360627, 644199, 1162296, 2083572, 3768866, 6777314, 12289160, 22158106, 40255496, 72765144, 132453122, 239936528, 437445448
Offset: 1
Cf.
A000992,
A001190,
A007317,
A063834,
A099323,
A196545,
A220418,
A273866,
A273873,
A289501,
A290261,
A300352,
A300442,
A300443,
A300862,
A300863,
A300864,
A300866.
-
a[n_]:=a[n]=(-1)^(n-1)+Sum[a[k]*a[n-k],{k,1,n/2}];
Array[a,50]
A158616
Table of expansion coefficients [x^m] of the Rayleigh polynomial of index 2n.
Original entry on oeis.org
1, 1, 2, 11, 5, 38, 14, 946, 1026, 362, 42, 4580, 4324, 1316, 132, 202738, 311387, 185430, 53752, 7640, 429, 3786092, 6425694, 4434158, 1596148, 317136, 33134, 1430, 261868876, 579783114, 547167306, 287834558, 92481350, 18631334, 2305702
Offset: 1
The polynomials of low index are Phi(2,x)=Phi(4,x) = 1 ; Phi(6,x)=2 ; Phi(8,x)=11+5x ; Phi(10,x)=38+14x ; Phi(12,x)=946+1026x+362x^2+42x^3 ;
Triangle begins:
1,
1,
2,
11,5,
38,14,
946,1026,362,42,
4580,4324,1316,132,
202738,311387,185430,53752,7640,429,
...
- Matthew House, Table of n, a(n) for n = 1..10015 (rows 1..93)
- Nand Kishore, The Rayleigh Polynomial, Proc. AMS 15 (6) (1964) 911-917.
- Nand Kishore, The Rayleigh Function, Proc. AMS 14 (4) (1963) 527-533.
- D. H. Lehmer, Zeros of the Bessel function J_{nu}(x), Math. Comp. 1 (1945), 405-407. Gives first 12 rows.
- D. H. Lehmer, Zeros of the Bessel function J_{nu}(x), Math. Comp., 1 (1943-1945), 405-407. Gives first 12 rows. [Annotated scanned copy]
-
sig2n := proc(n,nu) option remember ; if n = 1 then 1/4/(nu+1) ; else add( procname(k,nu)*procname(n-k,nu),k=1..n-1)/(nu+n) ; simplify(%) ; fi; end:
Phi2n := proc(n,nu) local k ; 4^n*mul( (nu+k)^(floor(n/k)),k=1..n)*sig2n(n,nu) ; factor(%) ; end:
for n from 1 to 14 do rpoly := Phi2n(n,nu) ; print(coeffs(rpoly)) ; od:
-
sig2n[n_, nu_] := sig2n[n, nu] = If[n == 1, 1/4/(nu + 1), Sum[sig2n[k, nu]*sig2n[n - k, nu], {k, 1, n - 1}]/(nu + n)] // Simplify;
Phi2n[n_, nu_] := 4^n*Product[(nu + k)^Floor[n/k], {k, 1, n}]*sig2n[n, nu];
T[n_] := CoefficientList[Phi2n[n, x], x];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 01 2023, after R. J. Mathar *)
Comments