Real-rooted atlas

A357367 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 2\,x + 7\,n x - 7\,n^{2} x + n^{2} x^{2} - 2\,n^{3} x^{2} + n^{4} x^{2}\right)\,P_{n}(x) = \left(-4 + 2\,x + 4\,n + 23\,n x + 8\,n x^{2} + 2\,n x^{3} + 14\,n^{2} x - n^{2} x^{2} - 7\,n^{3} x - 21\,n^{3} x^{2} - 4\,n^{3} x^{3} - 3\,n^{4} x^{2} + n^{5} x^{2} + 2\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(8\,x + 14\,x^{2} + 8\,x^{3} + 2\,x^{4} - 14\,n^{2} x^{2} - 18\,n^{2} x^{3} - 4\,n^{2} x^{4} + 2\,n^{4} x^{3} + 2\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1088
2024108132
3096108019203096
404809720403204200092520
50288090720645120151200010886403339360
60201609072009676800378000005987520032598720140878080
7016128097977601451520008316000002155507200254270016011070259206791944320
8014515201143072002235340800174636000006538371840012459230784011623772160042032390400368060837760
90145152001437004800357654528003632428800001830744115200498369231360074392141824005716405094400176432256000022134838118400
10015966720019399564800597793996800762810048000049430091110400179412923289600379399923302400463028812646400301699157760000811235513088001462339913126400
11019160064002802159360001046139494400016345929600000013181357629440006100039391846400170729965486080002932515813427200030169915776000000170359457748480004054836680294400105251230891699200
12024908083200431532541440019179224064000035961045120000003521305538150400020130129993093120071365125573181440016128836973849600002323083514752000000206134943875660800010258736801144832002189048635597824008196053042598220800
13034871316480070614415872000368241102028800081511702272000000950752495300608000655667091203604480028546050229272576000812893383482019840001533235119736320000001896441483656079360001477258099364855808006567145906793472000012693891496366080000686486897901258854400
1405230697472000122398320844800073982985043968000190737383316480000026092874037694464000213091804641171456000111329595894163046400038747917945976279040009168746016023193600000147922435725174190080001600362940978593792000011098476582480967680000445555591522449408000078689174326963200000061534798257551883264000
Technical details
Node id
n-deaf79c588bffa6b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-ccba683caf91532c

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 0,        8;
  2: 0,       24,        108;
  3: 0,       96,       1080,        1920;
  4: 0,      480,       9720,       40320,        42000;
  5: 0,     2880,      90720,      645120,      1512000,       1088640;
  6: 0,    20160,     907200,     9676800,     37800000,      59875200,      32598720;
  7: 0,   161280,    9797760,   145152000,    831600000,    2155507200,    2542700160,    1107025920;
  8: 0,  1451520,  114307200,  2235340800,  17463600000,   65383718400,  124592307840,  116237721600,   42032390400;
  9: 0, 14515200, 1437004800, 35765452800, 363242880000, 1830744115200, 4983692313600, 7439214182400, 5716405094400, 1764322560000;
%F Formula
The row polynomials satisfy (4-(7*n^2-7*n+2)*x+(n^4-2*n^3+n^2)*x^2)*P(n,x) = (4*n-4-(7*n^3-14*n^2-23*n-2)*x+(n^5-3*n^4-21*n^3-n^2+8*n)*x^2+(2*n^5-4*n^3+2*n)*x^3)*P(n-1,x) + (8*x-(14*n^2-14)*x^2+(2*n^4-18*n^2+8)*x^3+(2*n^4-4*n^2+2)*x^4)*d/dx 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 - (7*n^3 - 14*n^2 - 23*n - 2)*x + (n^5 - 3*n^4 - 21*n^3 - n^2 + 8*n)*x^2 + (2*n^5 - 4*n^3 + 2*n)*x^3)*P[n - 1, x] + (8*x - (14*n^2 - 14)*x^2 + (2*n^4 - 18*n^2 + 8)*x^3 + (2*n^4 - 4*n^2 + 2)*x^4)*D[P[n - 1, x], x])/(4 - (7*n^2 - 7*n + 2)*x + (n^4 - 2*n^3 + n^2)*x^2)]];
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 - (7*n^3 - 14*n^2 - 23*n - 2)*x + (n^5 - 3*n^4 - 21*n^3 - n^2 + 8*n)*x^2 + (2*n^5 - 4*n^3 + 2*n)*x^3)*v[i - 1] + (8*x - (14*n^2 - 14)*x^2 + (2*n^4 - 18*n^2 + 8)*x^3 + (2*n^4 - 4*n^2 + 2)*x^4)*deriv(v[i - 1], x))/(4 - (7*n^2 - 7*n + 2)*x + (n^4 - 2*n^3 + n^2)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.