Real-rooted atlas

A038336 → 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

\(\left(11 + 10\,x + 5\,n x\right)\,P_{n}(x) = \left(242 + 440\,x + 150\,x^{2} + 110\,n x + 50\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 484 + 880\,x + 300\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
04848803001664
1106482904019800400063488
2234256851840871200352000500002359296
35153632234256003194400019360000550000060000085983232
411337990461843584010541520008518400003630000007920000070000003087007744
52494357888158731865603246788160032795840000186340000006098400000107800000080000000109521666048
654875873536399097262080952391193600115441356800081989600000035777280000094864000000140800000009000000003848290697216
712072692177929877657236480269390651904003809564774400032467881600000177097536000006261024000000139392000000017820000000010000000000134140418588672
826559922791424241453843558400740824292736000119729178624000011904889920000007792291584000003443563200000001022208000000001960200000000022000000000001100000000004644337115725824
95843183014113285843183014113280199199420935680003621807653376000041156905152000000314289093888000001666684588800000061843584000000001581228000000000266200000000000266200000000001200000000000159877786771652608
1012855002631049216140236392338718720525886471270195200106239691165696000013581778700160000001185318868377600000733341219072000000326534123520000000104361048000000000234256000000000003513840000000000316800000000000130000000000005476377146882523136
11282810057883082752334230068407279616013673048253025075200303845516733890560004315987453606400000042375149544499200000299622269506560000001556479322112000000059694519456000000001674930400000000000334986080000000000453024000000000003718000000000000140000000000000186773283746309210112
126221821273427820544791868162072631705603509415718276435968008507674468548935680001329324135710771200000145017178441175040000011535457376002560000006848509017292800000003064318665408000000001031757126400000000002579392816000000000046510464000000000005725720000000000004312000000000000015000000000000006345679961356085755904
13136880068015412051968186654638202834616320089085168233171066880002339610478850957312000039879724071323136000000478556688855877632000004229667704534272000000028250099696332800000000144460737083520000000005674664195200000000000170239925856000000000038371132800000000000062982920000000000000711480000000000000049500000000000000016000000000000000214867674970568857223168
143011361496339065143296438016217649318566297602239855658434015395840006334945296581053644800001169805239425478656000000153138140433880842240000014888430319960637440000001104892788123238400000000635627243167488000000000285354542387200000000000998740898355200000000002701327749120000000000055424969600000000000008348032000000000000008712000000000000000056320000000000000001700000000000000007253554917687775048237056
Technical details
Node id
n-627115c3d1da65c2 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-0df1d6c1b80d1c27

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:             484,              880,               300;
  1:           10648,            29040,             19800,              4000;
  2:          234256,           851840,            871200,            352000,             50000;
  3:         5153632,         23425600,          31944000,          19360000,           5500000,            600000;
  4:       113379904,        618435840,        1054152000,         851840000,         363000000,          79200000,           7000000;
  5:      2494357888,      15873186560,       32467881600,       32795840000,       18634000000,        6098400000,        1078000000,         80000000;
  6:     54875873536,     399097262080,      952391193600,     1154413568000,      819896000000,      357772800000,       94864000000,      14080000000,        900000000;
  7:   1207269217792,    9877657236480,    26939065190400,    38095647744000,    32467881600000,    17709753600000,     6261024000000,    1393920000000,     178200000000,     10000000000;
  8:  26559922791424,  241453843558400,   740824292736000,  1197291786240000,  1190488992000000,   779229158400000,   344356320000000,  102220800000000,   19602000000000,   2200000000000,   110000000000;
  9: 584318301411328, 5843183014113280, 19919942093568000, 36218076533760000, 41156905152000000, 31428909388800000, 16666845888000000, 6184358400000000, 1581228000000000, 266200000000000, 26620000000000, 1200000000000;
%F Formula
The row polynomials satisfy (11+(5*n+10)*x)*P(n,x) = (242+(110*n+440)*x+(50*n+150)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 880*x + 300*x^2.
%t Mathematica
P[0, x_] := 484 + 880*x + 300*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((242 + (110*n + 440)*x + (50*n + 150)*x^2)*P[n - 1, x])/(11 + (5*n + 10)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 484 + 880*x + 300*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((242 + (110*n + 440)*x + (50*n + 150)*x^2)*v[i - 1])/(11 + (5*n + 10)*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

Same fingerprint ?

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.