Real-rooted atlas

A038336 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 20\,x + 10\,n x\right)\,P_{n}(x) = \left(242 + 880\,x + 600\,x^{2} + 220\,n x + 200\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 484 + 1760\,x + 1200\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0484176012003444
110648580807920032000179928
22342561703680348480028160008000009038736
35153632468512001277760001548800008800000019200000441860832
41133799041236871680421660800068147200005808000000253440000044800000021171979584
52494357888317463731201298715264002623667200002981440000001951488000006899200000010240000000999003777408
6548758735367981945241603809564774400923530854400013118336000000114487296000006071296000000180224000000023040000000046568945316096
71207269217792197553144729601077562607616003047651819520005194861056000005667121152000004007055360000001784217600000004561920000000051200000000002149548743204352
82655992279142448290768711680029632971709440009578334289920000190478238720000002493533306880000022038804480000000130842624000000005018112000000000112640000000000011264000000000098414474891572224
95843183014113281168636602822656079679768374272000289744612270080000658510482432000000100572510044160000010666781368320000007915978752000000004047943680000000001362944000000000002725888000000000024576000000000004475011907879589888
101285500263104921628047278467743744021035458850807808008499175293255680000217308459202560000003793020378808320000046933838020608000000417963678105600000002671642828800000000011993907200000000000359817216000000000064880640000000000053248000000000000202297866553152147456
112828100578830827526684601368145592320546921930121003008002430764133871124480006905579925770240000001356004785423974400000191758252484198400000019922935323033600000001528179698073600000000857564364800000000000343025745920000000000927793152000000000001522892800000000000011468800000000000009099099784965183823872
12622182127342782054415837363241452634112014037662873105743872006806139574839148544000212691861713723392000004640549710117601280000073826927206416384000000876609154213478400000007844655783444480000000052825964871680000000000264129824358400000000009525343027200000000000234525491200000000000035323904000000000000024576000000000000000407470945337315053092864
1313688006801541205196837330927640566923264003563406729326842675200018716883830807658496000063807558514117017600000015313814043388084224000002706987330901934080000000361601276113059840000000036981948693381120000000002905428067942400000000000174325684076544000000000078584079974400000000000025797804032000000000000058284441600000000000000811008000000000000000052428800000000000000018176747387655880194490368
14301136149633906514329687603243529863713259520895942263373606158336000506795623726484291584000018716883830807658496000000490042049388418695168000009528595404774807961600000014142627687977451520000000016272057425087692800000000014610152570224640000000000010227106799157248000000000055323192301977600000000000227020675481600000000000006838707814400000000000000142737408000000000000000018454937600000000000000011141120000000000000000808068032988070182681378816
Technical details
Node id
n-b5b5c00898328314 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-d4975513c6489427

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,              1760,              1200;
  1:           10648,             58080,             79200,              32000;
  2:          234256,           1703680,           3484800,            2816000,             800000;
  3:         5153632,          46851200,         127776000,          154880000,           88000000,            19200000;
  4:       113379904,        1236871680,        4216608000,         6814720000,         5808000000,          2534400000,           448000000;
  5:      2494357888,       31746373120,      129871526400,       262366720000,       298144000000,        195148800000,         68992000000,        10240000000;
  6:     54875873536,      798194524160,     3809564774400,      9235308544000,     13118336000000,      11448729600000,       6071296000000,      1802240000000,       230400000000;
  7:   1207269217792,    19755314472960,   107756260761600,    304765181952000,    519486105600000,     566712115200000,     400705536000000,    178421760000000,     45619200000000,      5120000000000;
  8:  26559922791424,   482907687116800,  2963297170944000,   9578334289920000,  19047823872000000,   24935333068800000,   22038804480000000,  13084262400000000,   5018112000000000,   1126400000000000,   112640000000000;
  9: 584318301411328, 11686366028226560, 79679768374272000, 289744612270080000, 658510482432000000, 1005725100441600000, 1066678136832000000, 791597875200000000, 404794368000000000, 136294400000000000, 27258880000000000, 2457600000000000;
%F Formula
The row polynomials satisfy (11+(10*n+20)*x)*P(n,x) = (242+(220*n+880)*x+(200*n+600)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1760*x + 1200*x^2.
%t Mathematica
P[0, x_] := 484 + 1760*x + 1200*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((242 + (220*n + 880)*x + (200*n + 600)*x^2)*P[n - 1, x])/(11 + (10*n + 20)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 484 + 1760*x + 1200*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((242 + (220*n + 880)*x + (200*n + 600)*x^2)*v[i - 1])/(11 + (10*n + 20)*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.