Real-rooted atlas

A038336 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 + 275\,x + 100\,x^{2} + 165\,n x + 100\,n x^{2} + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2662 + 10890\,x + 9350\,x^{2} + 2250\,x^{3} + 3630\,n x + 4950\,n x^{2} + 1500\,n x^{3} + 550\,n^{2} x^{2} + 250\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 484 + 1760\,x + 900\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
048417609003144
110648580805940016000144128
22342561703680261360014080002500006209536
35153632468512009583200077440000275000003600000256376832
411337990412368716803162456000340736000018150000004752000004900000010259267584
52494357888317463731209740364480013118336000093170000000365904000007546000000640000000400774135808
6548758735367981945241602857173580800461765427200040994800000002146636800000664048000000112640000000810000000015358803050496
71207269217792197553144729608081719557120015238259097600016233940800000010625852160000043827168000000111513600000001603800000000100000000000579442627837952
8265599227914244829076871168002222472878208000478916714496000059524449600000004675374950400000241049424000000081776640000000017641800000000022000000000000121000000000021576816183476224
958431830141132811686366028226560597598262807040001448723061350400002057845257600000001885734563328000001166679212160000004947486720000000014231052000000000266200000000000029282000000000014400000000000794603859254181888
101285500263104921628047278467743744015776594138105856004249587646627840000679088935008000000071119132102656000005133388533504000000261227298816000000093924943200000000023425600000000000038652240000000000380160000000000016900000000000028985167201756512256
11282810057883082752668460136814559232041019144759075225600121538206693556224000215799372680320000000254250897266995200000209735588654592000000124518345768960000000537250675104000000001674930400000000000036848468800000000005436288000000000004833400000000000019600000000000001048582108439927324672
126221821273427820544158373632414526341120105282471548293079040034030697874195742720006646620678553856000000870103070647050240000080748201632017920000005478807213834240000000275788679886720000000010317571264000000000002837332097600000000005581255680000000000074434360000000000006036800000000000002250000000000000037659028026478049624064
1313688006801541205196837330927640566923264002672555046995132006400093584419154038292480000199398620356615680000000287134013313526579200000296076739317399040000000226000797570662400000000130014663375168000000000567466419520000000000001872639184416000000000046045359360000000000008187779600000000000009960720000000000000074250000000000000002560000000000000001343808412281593416122368
143011361496339065143296876032435298637132595206719566975302046187520002533978118632421457920000584902619712739328000000091882884260328505344000001042190122397244620800000088391423049859072000000005720645188507392000000000285354542387200000000000010986149881907200000000003241593298944000000000007205246048000000000000011687244800000000000000130680000000000000000090112000000000000000289000000000000000047677012011051938077474816
Technical details
Node id
n-17ce7c9b5675d067 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-44310a99e9adba32

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:             484,              1760,               900;
  1:           10648,             58080,             59400,              16000;
  2:          234256,           1703680,           2613600,            1408000,             250000;
  3:         5153632,          46851200,          95832000,           77440000,           27500000,            3600000;
  4:       113379904,        1236871680,        3162456000,         3407360000,         1815000000,          475200000,           49000000;
  5:      2494357888,       31746373120,       97403644800,       131183360000,        93170000000,        36590400000,         7546000000,         640000000;
  6:     54875873536,      798194524160,     2857173580800,      4617654272000,      4099480000000,      2146636800000,       664048000000,      112640000000,        8100000000;
  7:   1207269217792,    19755314472960,    80817195571200,    152382590976000,    162339408000000,    106258521600000,     43827168000000,    11151360000000,     1603800000000,     100000000000;
  8:  26559922791424,   482907687116800,  2222472878208000,   4789167144960000,   5952444960000000,   4675374950400000,   2410494240000000,   817766400000000,   176418000000000,   22000000000000,   1210000000000;
  9: 584318301411328, 11686366028226560, 59759826280704000, 144872306135040000, 205784525760000000, 188573456332800000, 116667921216000000, 49474867200000000, 14231052000000000, 2662000000000000, 292820000000000, 14400000000000;
%F Formula
The row polynomials satisfy (121+(165*n+275)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (2662+(3630*n+10890)*x+(550*n^2+4950*n+9350)*x^2+(250*n^2+1500*n+2250)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 484 + 1760*x + 900*x^2.
%t Mathematica
P[0, x_] := 484 + 1760*x + 900*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2662 + (3630*n + 10890)*x + (550*n^2 + 4950*n + 9350)*x^2 + (250*n^2 + 1500*n + 2250)*x^3)*P[n - 1, x])/(121 + (165*n + 275)*x + (25*n^2 + 100*n + 100)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 484 + 1760*x + 900*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2662 + (3630*n + 10890)*x + (550*n^2 + 4950*n + 9350)*x^2 + (250*n^2 + 1500*n + 2250)*x^3)*v[i - 1])/(121 + (165*n + 275)*x + (25*n^2 + 100*n + 100)*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

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.