Real-rooted atlas

A038301 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(20 + 22\,x + 11\,n x\right)\,P_{n}(x) = \left(400 + 880\,x + 363\,x^{2} + 220\,n x + 121\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 880\,x + 363\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
04008803631643
180002640021780532461504
2160000704000871200425920732052234325
33200000176000002904000021296000732050096630679422806
464000000422400000871200000851840000439230000115956720124009272777027647
5128000000098560000002439360000029814400000204974000008116970400173612978015589736895850397548
6256000000002252800000006504960000009540608000008198960000004329050880001388903824002494357888019292299293274001079209
75120000000005068800000000167270400000002862182400000029516256000000194807289600008333422944000224492209920034726138722023579476910110875834867330
8102400000000001126400000000004181760000000008177664000000009838752000000007792291584000004166711472000001496614732800003472613872200047158953820002853116706113727986724654611
92048000000000002478080000000000102220800000000002248857600000000030921792000000000285717358080000001833353047680000082313810304000002546583506280000518748492020000627685675344203423740047332124583499621081752
10409600000000000054067200000000000245329920000000000599695360000000000927653760000000000979602370560000000733341219072000000395106289459200000152795010376800000414998793616000007532228104130400821697611359680407995688973734141581734113987453
11819200000000000001171456000000000000579870720000000000015592079360000000000267988864000000000003183707704320000000027238388136960000000171212725432320000007945340539593600000269749215850400000065279310235796800010682068947675840010607887913316980483317970015034137053324379207658414
12163840000000000000025231360000000000000135303168000000000000396889292800000000000750368819200000000000990486841344000000000953343584793600000000684850901729280000000370782558514368000000151059560876224000000456955171650577600009969931017830784000148510430786437720013532903160420952056962475037486154517246065027332879335
13327680000000000000005406720000000000000003122380800000000000000992223232000000000000020464604160000000000000297146052403200000000003177811949312000000000025681908814848000000000158906810791872000000007552978043811200000000274173102990346560000074774482633730880000014851043078643772000020299354740631428000170887425112458450066835970710650416148361011303276078782916
146553600000000000000001153433600000000000000071368704000000000000000244239564800000000000000545722777600000000000000864424879718400000000000101689982377984000000000091313453563904000000000063562724316748800000000034527899628851200000000014622565492818483200000047855668885587763200000118808344629150176000002165264505667352320000273419880179933520000213875106274081331207811454076807267374857309232311849987499857
Technical details
Node id
n-f268fc4a434ae144 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-958257b1df4e6ac9

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             400,              880,               363;
  1:            8000,            26400,             21780,              5324;
  2:          160000,           704000,            871200,            425920,             73205;
  3:         3200000,         17600000,          29040000,          21296000,           7320500,            966306;
  4:        64000000,        422400000,         871200000,         851840000,         439230000,         115956720,          12400927;
  5:      1280000000,       9856000000,       24393600000,       29814400000,       20497400000,        8116970400,        1736129780,        155897368;
  6:     25600000000,     225280000000,      650496000000,      954060800000,      819896000000,      432905088000,      138890382400,      24943578880,       1929229929;
  7:    512000000000,    5068800000000,    16727040000000,    28621824000000,    29516256000000,    19480728960000,     8333422944000,    2244922099200,     347261387220,     23579476910;
  8:  10240000000000,  112640000000000,   418176000000000,   817766400000000,   983875200000000,   779229158400000,   416671147200000,  149661473280000,   34726138722000,   4715895382000,   285311670611;
  9: 204800000000000, 2478080000000000, 10222080000000000, 22488576000000000, 30921792000000000, 28571735808000000, 18333530476800000, 8231381030400000, 2546583506280000, 518748492020000, 62768567534420, 3423740047332;
%F Formula
The row polynomials satisfy (20+(11*n+22)*x)*P(n,x) = (400+(220*n+880)*x+(121*n+363)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 880*x + 363*x^2.
%t Mathematica
P[0, x_] := 400 + 880*x + 363*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((400 + (220*n + 880)*x + (121*n + 363)*x^2)*P[n - 1, x])/(20 + (11*n + 22)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 400 + 880*x + 363*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((400 + (220*n + 880)*x + (121*n + 363)*x^2)*v[i - 1])/(20 + (11*n + 22)*x)); 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

Proof chains ? JSON

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.