Real-rooted atlas

A034870 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 - x + 2\,n x\right)\,P_{n}(x) = \left(8 + 12\,x + 6\,x^{2} + x^{3} + 8\,n x + 8\,n x^{2} + 2\,n x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
148315
2166472325189
3643847206403007272187
425620485376716856002688784128924057
510241024034560614406720048384235207680162020011255879
640964915220275245056063360060825641395220275271280176002904288132657205
71638422937611182082981888512512061501445381376351436817297286406401761763494447323921527103491
86553610485765898240183500803727360053673984574013444685824029652480146432005637632167731237856062720720051217272629233
92621444718592300810241069547522506752004211343365322670085214044164032737282489344001232225284888166415445248383846473440010444810404648192711943423
10104857620971520149422080597688320158760960030482104324445306880508035072046437580803439820800208109158410319462404192281601389158403720960079380481317840164160144408002126732013741
114194304922746887266631683229614080958791680020709900288342288629764470708633647152005120407486464002913528217617336696832860815155235655065601227916800349274112811789441516838422237602464001940496823261508830075
1216777216402653184347288371216978542592557108428801337060229122469847367683629163479044337984471044284429107203534654013442453809397761439908986887156944076830124892160107110727683200769792797423616163668736272025603570336356224253921152252541865828329
136710886417448304641635778560087241523200313524224000827703951360168989556736027590131712003685869158400409541017600038292085145603038049730560205701283840011928240128005933690880002531708108809246668288028795852800759890560016839680003094291204630912055016004992003250013522724571369673847
142684354567516192768761014517764396972769281717567488000494659436544011061134622720198648948326402933175877632036212047872000378415900262403377596465152025918361763840171766657843209859055616000490761879552021183276441607916241715202553232281607072665600016709172480333425664055456128075479040819000068140840824156829236393522034597
Technical details
Node id
n-b8dd0525caf58a39 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-8f73d974446259d9

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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,       8,        3;
  2:     16,      64,       72,        32,         5;
  3:     64,     384,      720,       640,       300,        72,         7;
  4:    256,    2048,     5376,      7168,      5600,      2688,       784,       128,         9;
  5:   1024,   10240,    34560,     61440,     67200,     48384,     23520,      7680,      1620,       200,        11;
  6:   4096,   49152,   202752,    450560,    633600,    608256,    413952,    202752,     71280,     17600,      2904,      288,       13;
  7:  16384,  229376,  1118208,   2981888,   5125120,   6150144,   5381376,   3514368,   1729728,    640640,    176176,    34944,     4732,     392,     15;
  8:  65536, 1048576,  5898240,  18350080,  37273600,  53673984,  57401344,  46858240,  29652480,  14643200,   5637632,  1677312,   378560,   62720,   7200,    512,    17;
  9: 262144, 4718592, 30081024, 106954752, 250675200, 421134336, 532267008, 521404416, 403273728, 248934400, 123222528, 48881664, 15445248, 3838464, 734400, 104448, 10404, 648, 19;
%F Formula
The row polynomials satisfy (2+(2*n-1)*x)*P(n,x) = (8+(8*n+12)*x+(8*n+6)*x^2+(2*n+1)*x^3)*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[((8 + (8*n + 12)*x + (8*n + 6)*x^2 + (2*n + 1)*x^3)*P[n - 1, x])/(2 + (2*n - 1)*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] = ((8 + (8*n + 12)*x + (8*n + 6)*x^2 + (2*n + 1)*x^3)*v[i - 1])/(2 + (2*n - 1)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.