Real-rooted atlas

A038336 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(21 + 10\,x + 10\,n\right)\,P_{n}(x) = \left(682 + 840\,x + 200\,x^{2} + 220\,n + 200\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1364 + 1680\,x + 400\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0136416804003444
13968881840504008000179928
210860963175040327360013440001600009038736
328579232108609600158752000109120000336000003200000441860832
473181574434295078406516576000635008000032736000008064000006400000021171979584
51836754444810245420416024006554880030410688000022225280000091660800000188160000001280000000999003777408
6453973135616293880711168081963363328001280349593600012164275200000711208960000024442880000004300800000002560000000046568945316096
711084926454272817151644108802644926400512004917801799680005761573171200004379139072000002133626880000006285312000000096768000000005120000000002149548743204352
82680137663498242216985290854400817151644108800017632842670080000245890089984000002304629268480000014597130240000000609607680000000015713280000000002150400000000001024000000000098414474891572224
964275013155246085896302859696128024386838199398400059924453901312000096980634685440000010819163959296000008450307317760000004587669504000000001676421120000000003841024000000000047308800000000002048000000000004475011907879589888
101530913949697679361542600315725905920707556343163535360019509470559518720000359546723407872000004655070464901120000043276655837184000000289724822323200000001376300851200000000044704563200000000009218457600000000001032192000000000004096000000000000202297866553152147456
113625110741955878912398037626921396633602005380410443677696006132154974083973120001268115586368716800000186964296172093440000020171972014571520000001607418645381120000000941605672550400000000397598023680000000000116231864320000000000217890816000000000002236416000000000000819200000000000009099099784965183823872
12854086374806909911041015031007747646095360557252677689955287040018716883830807658496000429250848185878118400007101447283664814080000087250004880310272000000806878880582860800000005625965258833920000000029294398701568000000000111327446630400000000002958629273600000000000508411904000000000000481689600000000000001638400000000000000407470945337315053092864
13200342645004375821516825622591244207297331200152254651162146914304000557252677689955287040000140376628731057438720000025755050891152687104000003550723641832407040000000373928592344186880000000030257958021857280000000001875321752944640000000000878831961047040000000000303620308992000000000000739657318400000000000001173258240000000000000010321920000000000000003276800000000000000018176747387655880194490368
1446812983261270921773056641096464014002628853760409961459907316757299200016240496123962337525760000445802142151964229632000008984104238787676078080000013736027141948099788800000016231879505519575040000000014957143693767475200000000010758385074438144000000000060010296094228480000000000255660206850048000000000008096541573120000000000000182069493760000000000000026817331200000000000000022020096000000000000000655360000000000000000808068032988070182681378816
Technical details
Node id
n-66e422e8d13851fe · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3e3a8419375cf2f4

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:             1364,              1680,                400;
  1:            39688,             81840,              50400,               8000;
  2:          1086096,           3175040,            3273600,            1344000,             160000;
  3:         28579232,         108609600,          158752000,          109120000,           33600000,             3200000;
  4:        731815744,        3429507840,         6516576000,         6350080000,         3273600000,           806400000,           64000000;
  5:      18367544448,      102454204160,       240065548800,       304106880000,       222252800000,         91660800000,        18816000000,         1280000000;
  6:     453973135616,     2938807111680,      8196336332800,     12803495936000,     12164275200000,       7112089600000,      2444288000000,       430080000000,        25600000000;
  7:   11084926454272,    81715164410880,    264492640051200,    491780179968000,    576157317120000,     437913907200000,    213362688000000,     62853120000000,      9676800000000,      512000000000;
  8:  268013766349824,  2216985290854400,   8171516441088000,  17632842670080000,  24589008998400000,   23046292684800000,  14597130240000000,   6096076800000000,   1571328000000000,   215040000000000,   10240000000000;
  9: 6427501315524608, 58963028596961280, 243868381993984000, 599244539013120000, 969806346854400000, 1081916395929600000, 845030731776000000, 458766950400000000, 167642112000000000, 38410240000000000, 4730880000000000, 204800000000000;
%F Formula
The row polynomials satisfy (10*n+21+10*x)*P(n,x) = (220*n+682+(200*n+840)*x+200*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1364 + 1680*x + 400*x^2.
%t Mathematica
P[0, x_] := 1364 + 1680*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((220*n + 682 + (200*n + 840)*x + 200*x^2)*P[n - 1, x])/(10*n + 21 + 10*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1364 + 1680*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((220*n + 682 + (200*n + 840)*x + 200*x^2)*v[i - 1])/(10*n + 21 + 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.