Real-rooted atlas

A038301 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(10 + 22\,x + 11\,n x\right)\,P_{n}(x) = \left(200 + 880\,x + 726\,x^{2} + 220\,n x + 242\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 400 + 1760\,x + 1452\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0400176014523612
18000528008712042592190512
216000014080003484800340736011712809631440
332000003520000011616000017036800011712800030921792472977792
464000000844800000348480000068147200007027680000371061504079365932822740274368
512800000001971200000097574400000238515200000327958400000259743052800111112305920199548631041075850221824
6256000000004505600000002601984000000763248640000013118336000000138529628160008888984473600319277809664049388286182450257574648064
7512000000000101376000000006690816000000022897459200000047226009600000062338332672000053333906841600028735002869760088898915128320120726921779202323836479139840
810240000000000225280000000000167270400000000065421312000000001574200320000000024935333068800000266669534208000001915666857984000088898915128320002414538435584000292159150705664106547902568561664
9204800000000000495616000000000040888320000000000179908608000000000494748672000000000914295545856000000117334595051520000010536167718912000006519253776076800002655992279142400006427501315524608070118196169359364850776266556502016
104096000000000000108134400000000000981319680000000000479756288000000000014842460160000000000313472758579200000004693383802060800000050573605050777600000391155226564608000002124793823313920000077130015786295296001682836708064624640167115034203639808219514706259803394048
1181920000000000000234291200000000000023194828800000000000124736634880000000000428782182400000000000101878646538240000000017432568407654400000002191522885533696000000203400717813596160000013811159851540480000006684601368145592320002187687720484012032004344990889294635008039593408103631585289882465991617815543808
12163840000000000000050462720000000000000541212672000000000000317511434240000000000012005901107200000000000316955789230080000000006101398942679040000000087660915421347840000000949203349796782080000007734249516862668800000046792209577019146240000204184187245174456320006082987245012489011200110861542690168438784093327319101417308160442903201453603318579200
13327680000000000000001081344000000000000000124895232000000000000007937785856000000000000032743366656000000000000095086736769024000000000020337996475596800000000003287284328300544000000000406801435627192320000000038671247584313344000000002807532574621148774400000153138140433880842240000060829872450124890112000016629231403525265817600027998195730425192448000219008108824659283148819771198912888852141375488
1465536000000000000000023068672000000000000000285474816000000000000000195391651840000000000000087315644416000000000000002766159615098880000000000065081588721909760000000000116881220561797120000000000162720574250876928000000000176782846099718144000000000149735070646461267968000000980084098776837390336000004866389796009991208960000017737846830426950205440000447971131686803079168000070082594823890970607616051193145437764107436032879499461318507325901832192
Technical details
Node id
n-d9a911fe9fa9ccfa · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5938108d9525e299

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,             1760,              1452;
  1:            8000,            52800,             87120,              42592;
  2:          160000,          1408000,           3484800,            3407360,            1171280;
  3:         3200000,         35200000,         116160000,          170368000,          117128000,           30921792;
  4:        64000000,        844800000,        3484800000,         6814720000,         7027680000,         3710615040,           793659328;
  5:      1280000000,      19712000000,       97574400000,       238515200000,       327958400000,       259743052800,        111112305920,         19954863104;
  6:     25600000000,     450560000000,     2601984000000,      7632486400000,     13118336000000,     13852962816000,       8888984473600,       3192778096640,       493882861824;
  7:    512000000000,   10137600000000,    66908160000000,    228974592000000,    472260096000000,    623383326720000,     533339068416000,     287350028697600,     88898915128320,     12072692177920;
  8:  10240000000000,  225280000000000,  1672704000000000,   6542131200000000,  15742003200000000,  24935333068800000,   26666953420800000,   19156668579840000,   8889891512832000,   2414538435584000,   292159150705664;
  9: 204800000000000, 4956160000000000, 40888320000000000, 179908608000000000, 494748672000000000, 914295545856000000, 1173345950515200000, 1053616771891200000, 651925377607680000, 265599227914240000, 64275013155246080, 7011819616935936;
%F Formula
The row polynomials satisfy (10+(11*n+22)*x)*P(n,x) = (200+(220*n+880)*x+(242*n+726)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 400 + 1760*x + 1452*x^2.
%t Mathematica
P[0, x_] := 400 + 1760*x + 1452*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((200 + (220*n + 880)*x + (242*n + 726)*x^2)*P[n - 1, x])/(10 + (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 + 1760*x + 1452*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((200 + (220*n + 880)*x + (242*n + 726)*x^2)*v[i - 1])/(10 + (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.