Real-rooted atlas

A038337 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(529 + 1265\,x + 484\,x^{2} + 759\,n x + 484\,n x^{2} + 121\,n^{2} x^{2}\right)\,P_{n}(x) = \left(12167 + 52371\,x + 47311\,x^{2} + 11979\,x^{3} + 17457\,n x + 25047\,n x^{2} + 7986\,n x^{3} + 2783\,n^{2} x^{2} + 1331\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 529 + 2024\,x + 1089\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0529202410893642
112167698287514121296178432
22798412141392345648619592323660258202976
3643634361565020132498630112655840420928755797836361046544
414803588916991945524571202735518216864029044083758001013688680648915391918048
534048254474559505381214719272806720858228776015586991612564408160124139758447291247178944640275995008
6783109852811198498557344451391032738876758281895687170016141750395036715427212857777150682294809256961736306936126119553065728
7180115266146331011150171276133482776824188264816072540096296838668268450204431500233576887186623396922375127580953635941553577272357947691001048681208975104
841426511213649792507171043720383762983369540587010995263174401137881561695725094038490107444965101323084532290182093114539776041332786613860554232796893000313842837672141548280991310336
995280975791392720050431427406116107880038658326385275172272519789040411262907298597750396528966619726248258126948077333874115173894946408320348573167110223556860448806964500794022379310413410848805679841627701142083366912
102191462443202032150308355217855345629774890669698082268438616357273530560141885703018016223751563457068386349206411873839611557358204635759900104173926424051548530605424956311612902407340001095750883448369941133942703676358453039439566584963153444252196667392
1150403636193646738312535165175115623612809335664567247872342523146290824785637444713758355820761211255843420793093980158925071825776936643004283168203502185800066561438282602130204412014717930644549486650010920983805035420402169524434199615580815858792430408885167664515802104762429797389061405069312
12115928363245387498093104863989528639079282171717366588781790779738593732405073613868815178301905742851100225209064610597362401241362041409875216998809222714573736110054680306176771878329809876367778730383473350898694026008791391963053513423611819562260409207233922553265581295830501121787974088277732728544371255622922592798354706089927663616
13266635235464391245607765127197419557487394057634037805625362909135212345698066458663987280476046741589207602688875721272906560900284282692782540452166516210202035628492369746108088203840380425748263439066976645174704971767667490649500606606045450692426229091569372449602941239256029362554184902050762653758425530227908894202947808083189908262510693755313704066563521077807644557656489984
14613261041568099864896118771120576693143690732815149518508907238250401206011016683727445257486080145987667420690331491255002412985723767375496509369628797488639727796535434888256983546740630862732236801749958442011819708092567091844899443573766512880003720517078764246880871752115505812290776475209241627013549850109886701638046103353170795682435520542396687306943120300039353019554430964940813279471930572354529132825998838694476668796928
Technical details
Node id
n-cacee0d6a2dd1bbb · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-85b48dabbb345501

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             529,              2024,               1089;
  1:           12167,             69828,              75141,              21296;
  2:          279841,           2141392,            3456486,            1959232,             366025;
  3:         6436343,          61565020,          132498630,          112655840,           42092875,            5797836;
  4:       148035889,        1699194552,         4571202735,         5182168640,         2904408375,          800101368,           86806489;
  5:      3404825447,       45595053812,       147192728067,       208582287760,       155869916125,        64408160124,        13975844729,         1247178944;
  6:     78310985281,     1198498557344,      4513910327388,      7675828189568,      7170016141750,      3950367154272,      1285777715068,       229480925696,       17363069361;
  7:   1801152661463,    31011150171276,    133482776824188,    264816072540096,    296838668268450,    204431500233576,     88718662339692,     23751275809536,     3594155357727,     235794769100;
  8:  41426511213649,   792507171043720,   3837629833695405,   8701099526317440,  11378815616957250,   9403849010744496,   5101323084532290,   1820931145397760,   413327866138605,   54232796893000,   3138428376721;
  9: 952809757913927, 20050431427406116, 107880038658326385, 275172272519789040, 411262907298597750, 396528966619726248, 258126948077333874, 115173894946408320, 34857316711022355, 6860448806964500, 794022379310413, 41084880567984;
%F Formula
The row polynomials satisfy (529+(759*n+1265)*x+(121*n^2+484*n+484)*x^2)*P(n,x) = (12167+(17457*n+52371)*x+(2783*n^2+25047*n+47311)*x^2+(1331*n^2+7986*n+11979)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 2024*x + 1089*x^2.
%t Mathematica
P[0, x_] := 529 + 2024*x + 1089*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12167 + (17457*n + 52371)*x + (2783*n^2 + 25047*n + 47311)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*P[n - 1, x])/(529 + (759*n + 1265)*x + (121*n^2 + 484*n + 484)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 529 + 2024*x + 1089*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12167 + (17457*n + 52371)*x + (2783*n^2 + 25047*n + 47311)*x^2 + (1331*n^2 + 7986*n + 11979)*x^3)*v[i - 1])/(529 + (759*n + 1265)*x + (121*n^2 + 484*n + 484)*x^2)); 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.