Real-rooted atlas

A156992 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + 3\,n x\right)\,P_{n}(x) = \left(4 + 12\,x + 9\,x^{2} + 4\,n + 18\,n x + 18\,n x^{2} + 6\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
141216
224144162330
31921728388825928400
41920230407776010368048600255000
52304034560015552003110400291600010497609000000
6322560580608032659200870912001224720008817984025719120362250000
75160960108380160731566080243855360045722880004938071040288054144070543872016380000000
8928972802229534720175575859207023034368016460236800023702740992020739898368010158317568021427701120822150000000
91857945600501645312004514807808002106910310400592568524800010666233446400124439390208009142485811200385698620160071425670400045360000000000
104087480320012262440960001241572147200066217181184000217275125760000469314271641600684416646144000670448959488000424268482176000157136474880000259275183552002728687500000000
119809952768003237284413440036419449651200021851669790720008194376171520000206498279522304003613719891640320044249631326208000373356264314880002074201468416000068448648457728001018244357222400177754500000000000
12255058771968009182115790848001136286829117440075752455274496000319580670689280000920392331585126400187913434365296640027611769947553792002912178861656064000215716952715264000010677989159405568003176922394533888004302082409264640012466203750000000000
1371416456151040027852417898905600376007641635225600275738937199165440012925262681210880000418778510871232512009771498586995425280016751140434849300480021200662112856145920019630242697089024000012955960180078755840057819987580516761600156595799697232896001945864966652006400936485550000000000000
1421424936845312000899847347503104000131602674572328960001052821396578631680005428610326108569600001954299717399085056000513003675817259827200010050684260909580288000148404634789993021440001648940386555478016000013603758189082693632000809479826127234662400032885117936418908160008172632859938426880009381848946357888000075025262812500000000000
Technical details
Node id
n-cf28bf26a0e97b2b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-311883d116e971c2

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:          1;
  1:          4,          12;
  2:         24,         144,          162;
  3:        192,        1728,         3888,          2592;
  4:       1920,       23040,        77760,        103680,         48600;
  5:      23040,      345600,      1555200,       3110400,       2916000,        1049760;
  6:     322560,     5806080,     32659200,      87091200,     122472000,       88179840,       25719120;
  7:    5160960,   108380160,    731566080,    2438553600,    4572288000,     4938071040,     2880541440,     705438720;
  8:   92897280,  2229534720,  17557585920,   70230343680,  164602368000,   237027409920,   207398983680,  101583175680,   21427701120;
  9: 1857945600, 50164531200, 451480780800, 2106910310400, 5925685248000, 10666233446400, 12443939020800, 9142485811200, 3856986201600, 714256704000;
%F Formula
The row polynomials satisfy (2+3*n*x)*P(n,x) = (4*n+4+(6*n^2+18*n+12)*x+(9*n^2+18*n+9)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 4 + (6*n^2 + 18*n + 12)*x + (9*n^2 + 18*n + 9)*x^2)*P[n - 1, x])/(2 + 3*n*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n + 4 + (6*n^2 + 18*n + 12)*x + (9*n^2 + 18*n + 9)*x^2)*v[i - 1])/(2 + 3*n*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.