A154986 → θ+1 → x ↦ x/(x+1)
nonnegative coefficients
Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel
Recurrence
\(\left(3 + 6\,x + 3\,x^{2} - 3\,n - 6\,n x - 3\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(-12\,x^{2} - 48\,x^{3} - 72\,x^{4} - 48\,x^{5} - 12\,x^{6} + 40\,n x^{2} + 160\,n x^{3} + 240\,n x^{4} + 160\,n x^{5} + 40\,n x^{6} - 63\,n^{2} x^{2} - 252\,n^{2} x^{3} - 378\,n^{2} x^{4} - 252\,n^{2} x^{5} - 63\,n^{2} x^{6} + 59\,n^{3} x^{2} + 236\,n^{3} x^{3} + 354\,n^{3} x^{4} + 236\,n^{3} x^{5} + 59\,n^{3} x^{6} - 32\,n^{4} x^{2} - 128\,n^{4} x^{3} - 192\,n^{4} x^{4} - 128\,n^{4} x^{5} - 32\,n^{4} x^{6} + 9\,n^{5} x^{2} + 36\,n^{5} x^{3} + 54\,n^{5} x^{4} + 36\,n^{5} x^{5} + 9\,n^{5} x^{6} - n^{6} x^{2} - 4\,n^{6} x^{3} - 6\,n^{6} x^{4} - 4\,n^{6} x^{5} - n^{6} x^{6}\right)\,P_{n-4}(x) + \left(-4\,x - 24\,x^{2} - 48\,x^{3} - 40\,x^{4} - 12\,x^{5} + 10\,n x + 60\,n x^{2} + 120\,n x^{3} + 100\,n x^{4} + 30\,n x^{5} - 12\,n^{2} x - 68\,n^{2} x^{2} - 132\,n^{2} x^{3} - 108\,n^{2} x^{4} - 32\,n^{2} x^{5} + 8\,n^{3} x + 42\,n^{3} x^{2} + 78\,n^{3} x^{3} + 62\,n^{3} x^{4} + 18\,n^{3} x^{5} - 2\,n^{4} x - 10\,n^{4} x^{2} - 18\,n^{4} x^{3} - 14\,n^{4} x^{4} - 4\,n^{4} x^{5}\right)\,P_{n-3}(x) + \left(-1 - 6\,x - 15\,x^{2} - 16\,x^{3} - 6\,x^{4} + n - 3\,n x^{2} - 2\,n x^{3} - n^{2} + 6\,n^{2} x + 23\,n^{2} x^{2} + 24\,n^{2} x^{3} + 8\,n^{2} x^{4} - 8\,n^{3} x - 24\,n^{3} x^{2} - 24\,n^{3} x^{3} - 8\,n^{3} x^{4} + 2\,n^{4} x + 6\,n^{4} x^{2} + 6\,n^{4} x^{3} + 2\,n^{4} x^{4}\right)\,P_{n-2}(x) + \left(4 + 18\,x + 24\,x^{2} + 10\,x^{3} - 4\,n - 18\,n x - 24\,n x^{2} - 10\,n x^{3} + 2\,n^{2} + 8\,n^{2} x + 10\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 3\,x\)
\(P_{2}(x) = 1 + 10\,x + 12\,x^{2}\)
\(P_{3}(x) = 1 + 25\,x + 80\,x^{2} + 60\,x^{3}\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | x5 | x6 | x7 | x8 | x9 | x10 | x11 | x12 | x13 | x14 | Pn(1) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 1 | 1 | ||||||||||||||
| 1 | 1 | 3 | 4 | |||||||||||||
| 2 | 1 | 10 | 12 | 23 | ||||||||||||
| 3 | 1 | 25 | 80 | 60 | 166 | |||||||||||
| 4 | 1 | 52 | 360 | 664 | 360 | 1437 | ||||||||||
| 5 | 1 | 95 | 1312 | 4632 | 5928 | 2520 | 14488 | |||||||||
| 6 | 1 | 158 | 4012 | 25400 | 59088 | 57696 | 20160 | 166515 | ||||||||
| 7 | 1 | 245 | 10584 | 117260 | 453040 | 776016 | 611328 | 181440 | 2149914 | |||||||
| 8 | 1 | 360 | 24752 | 462864 | 2889360 | 7894080 | 10662912 | 7034112 | 1814400 | 30782841 | ||||||
| 9 | 1 | 507 | 52512 | 1581968 | 15871632 | 66376368 | 138768768 | 154252800 | 87477120 | 19958400 | 484340076 | |||||
| 10 | 1 | 690 | 102924 | 4763024 | 75715776 | 478687872 | 1480568832 | 2496640896 | 2355108480 | 1170904320 | 239500800 | 8301993615 | ||||
| 11 | 1 | 913 | 189024 | 12865820 | 315303296 | 3019513920 | 13542384384 | 32928362496 | 46374198912 | 37952046720 | 16794224640 | 3113510400 | 154052600526 | |||
| 12 | 1 | 1180 | 328856 | 31698920 | 1157883320 | 16740292288 | 108340958208 | 370868424192 | 739608493824 | 892947879936 | 645015536640 | 257124049920 | 43589145600 | 3075424692885 | ||
| 13 | 1 | 1495 | 546624 | 72240168 | 3800533880 | 81697748808 | 766811401856 | 3660535862784 | 10050317012736 | 16933195987200 | 17866522450944 | 11544626580480 | 4186513658880 | 653837184000 | 65747931209856 | |
| 14 | 1 | 1862 | 873964 | 154041992 | 11306158864 | 352927863392 | 4814890097088 | 32037420669312 | 119515741878528 | 272665805686272 | 397026608348160 | 371773340559360 | 217256256000000 | 72256549478400 | 10461394944000 | 1498172396601195 |
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Technical details
- Node id
n-43d14eb8a375afdd· revision 0 · JSON- Definition source
derived:mobius_x_over_x_plus_one- Fingerprint
f6-992e456fd58bab57
Observed on the stored rows ?
- Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
- Palindromic rows: no
- γ-positive (gamma_positive): no
- Nonnegative coefficients (
nonn): yes - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): no
Matches ?
No OEIS matches.
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1; 1: 1, 3; 2: 1, 10, 12; 3: 1, 25, 80, 60; 4: 1, 52, 360, 664, 360; 5: 1, 95, 1312, 4632, 5928, 2520; 6: 1, 158, 4012, 25400, 59088, 57696, 20160; 7: 1, 245, 10584, 117260, 453040, 776016, 611328, 181440; 8: 1, 360, 24752, 462864, 2889360, 7894080, 10662912, 7034112, 1814400; 9: 1, 507, 52512, 1581968, 15871632, 66376368, 138768768, 154252800, 87477120, 19958400;
%F FormulaThe row polynomials satisfy (n^2-3*n+3+(2*n^2-6*n+6)*x+(n^2-3*n+3)*x^2)*P(n,x) = -((n^6-9*n^5+32*n^4-59*n^3+63*n^2-40*n+12)*x^2+(4*n^6-36*n^5+128*n^4-236*n^3+252*n^2-160*n+48)*x^3+(6*n^6-54*n^5+192*n^4-354*n^3+378*n^2-240*n+72)*x^4+(4*n^6-36*n^5+128*n^4-236*n^3+252*n^2-160*n+48)*x^5+(n^6-9*n^5+32*n^4-59*n^3+63*n^2-40*n+12)*x^6)*P(n-4,x) - ((2*n^4-8*n^3+12*n^2-10*n+4)*x+(10*n^4-42*n^3+68*n^2-60*n+24)*x^2+(18*n^4-78*n^3+132*n^2-120*n+48)*x^3+(14*n^4-62*n^3+108*n^2-100*n+40)*x^4+(4*n^4-18*n^3+32*n^2-30*n+12)*x^5)*P(n-3,x) - (n^2-n+1-(2*n^4-8*n^3+6*n^2-6)*x-(6*n^4-24*n^3+23*n^2-3*n-15)*x^2-(6*n^4-24*n^3+24*n^2-2*n-16)*x^3-(2*n^4-8*n^3+8*n^2-6)*x^4)*P(n-2,x) + (2*n^2-4*n+4+(8*n^2-18*n+18)*x+(10*n^2-24*n+24)*x^2+(4*n^2-10*n+10)*x^3)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 3*x, P(2,x) = 1 + 10*x + 12*x^2, P(3,x) = 1 + 25*x + 80*x^2 + 60*x^3.
%t MathematicaP[0, x_] := 1;
P[1, x_] := 1 + 3*x;
P[2, x_] := 1 + 10*x + 12*x^2;
P[3, x_] := 1 + 25*x + 80*x^2 + 60*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-((n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2 + (4*n^6 - 36*n^5 + 128*n^4 - 236*n^3 + 252*n^2 - 160*n + 48)*x^3 + (6*n^6 - 54*n^5 + 192*n^4 - 354*n^3 + 378*n^2 - 240*n + 72)*x^4 + (4*n^6 - 36*n^5 + 128*n^4 - 236*n^3 + 252*n^2 - 160*n + 48)*x^5 + (n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^6)*P[n - 4, x] - ((2*n^4 - 8*n^3 + 12*n^2 - 10*n + 4)*x + (10*n^4 - 42*n^3 + 68*n^2 - 60*n + 24)*x^2 + (18*n^4 - 78*n^3 + 132*n^2 - 120*n + 48)*x^3 + (14*n^4 - 62*n^3 + 108*n^2 - 100*n + 40)*x^4 + (4*n^4 - 18*n^3 + 32*n^2 - 30*n + 12)*x^5)*P[n - 3, x] - (n^2 - n + 1 - (2*n^4 - 8*n^3 + 6*n^2 - 6)*x - (6*n^4 - 24*n^3 + 23*n^2 - 3*n - 15)*x^2 - (6*n^4 - 24*n^3 + 24*n^2 - 2*n - 16)*x^3 - (2*n^4 - 8*n^3 + 8*n^2 - 6)*x^4)*P[n - 2, x] + (2*n^2 - 4*n + 4 + (8*n^2 - 18*n + 18)*x + (10*n^2 - 24*n + 24)*x^2 + (4*n^2 - 10*n + 10)*x^3)*P[n - 1, x])/(n^2 - 3*n + 3 + (2*n^2 - 6*n + 6)*x + (n^2 - 3*n + 3)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 3*x; v[3] = 1 + 10*x + 12*x^2; v[4] = 1 + 25*x + 80*x^2 + 60*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-((n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^2 + (4*n^6 - 36*n^5 + 128*n^4 - 236*n^3 + 252*n^2 - 160*n + 48)*x^3 + (6*n^6 - 54*n^5 + 192*n^4 - 354*n^3 + 378*n^2 - 240*n + 72)*x^4 + (4*n^6 - 36*n^5 + 128*n^4 - 236*n^3 + 252*n^2 - 160*n + 48)*x^5 + (n^6 - 9*n^5 + 32*n^4 - 59*n^3 + 63*n^2 - 40*n + 12)*x^6)*v[i - 4] - ((2*n^4 - 8*n^3 + 12*n^2 - 10*n + 4)*x + (10*n^4 - 42*n^3 + 68*n^2 - 60*n + 24)*x^2 + (18*n^4 - 78*n^3 + 132*n^2 - 120*n + 48)*x^3 + (14*n^4 - 62*n^3 + 108*n^2 - 100*n + 40)*x^4 + (4*n^4 - 18*n^3 + 32*n^2 - 30*n + 12)*x^5)*v[i - 3] - (n^2 - n + 1 - (2*n^4 - 8*n^3 + 6*n^2 - 6)*x - (6*n^4 - 24*n^3 + 23*n^2 - 3*n - 15)*x^2 - (6*n^4 - 24*n^3 + 24*n^2 - 2*n - 16)*x^3 - (2*n^4 - 8*n^3 + 8*n^2 - 6)*x^4)*v[i - 2] + (2*n^2 - 4*n + 4 + (8*n^2 - 18*n + 18)*x + (10*n^2 - 24*n + 24)*x^2 + (4*n^2 - 10*n + 10)*x^3)*v[i - 1])/(n^2 - 3*n + 3 + (2*n^2 - 6*n + 6)*x + (n^2 - 3*n + 3)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
- Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
- Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
Proof chains ? JSON
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
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from
theta_plus_one of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read…?
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.?
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Liu–Wang theorem with a lag row [LW07, Thm 2.1] RealRooted ? step › direct check
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Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- ↑ Affine, reversal and Möbius substitutions (elementary) (above)
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from
Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read by rows.?
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Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
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from
theta_plus_one of Triangle T(n, k) = coefficients of p(x, n) where p(x,n) = (x+1)*p(x, n-1) + n*(n-1)*x*p(x, n-2), read…?
References
- [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
- [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
- [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.