Real-rooted atlas

A013612 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + 50\,x + 100\,x^{2} + 30\,n x + 100\,n x^{2} + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(24 + 540\,x + 2550\,x^{2} + 3375\,x^{3} + 180\,n x + 1350\,n x^{2} + 2250\,n x^{3} + 150\,n^{2} x^{2} + 375\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 36 + 720\,x + 2025\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
03672020252781
12166480364505400097146
2129651840437400129600012656253052161
37776388800437400019440000379687502733750089516826
4466562799360393660002332800006834375009841500005581406252501220141
5279936195955203306744002449440000956812500020667150000234419062501093500000067412171106
616796161343692802645395200235146240001148175000003306744000005626057500005248800000002075941406251766867858721
7100776969069926402040733440021163161600012400290000004464104400000101269035000001417176000000011210083593750384433593750045290172451986
860466176604661760015305500800018139852800001240029000000053569252800000151903552500000283435200000000336302507812500230660156250000697746972656251140018803999901
93627970563990767616011224033920001496537856000011691702000000058926178080000020051268930000004676680800000000739865517187500076117851562500004605130019531250124556484375000028265249737631466
10217678233626121388032080813044224001197230284800001052253180000000606097831680000024061522716000000673442035200000001331757930937500001826828437500000001657846807031250008968066875000000021927131103515625691898142856755681
11130606940161697890222080573037949952009338396221440009119527560000000590945385888000002681141102640000008754746457600000002077542372262500000356231545312500000043104016982812500003497546081250000000171031622607421875038145423339843750016752371740932261546
1278364164096109709829734404011265649664007131138932736000766040315040000005515490268288000002815198157772000000105056957491200000002908559321167500000059846899612500000000905184356639062500009793129027500000000071833281495117187500320421556054687500006568408355712890625401782654499449718061
134701849845767052774768640027770300651520005348354199552000062676025776000000049639412414592000002815198157772000000011818907717760000000037395762700725000000089770349418750000000016293318419503125000002203454031187500000000215499844485351562500014418970022460937500005911567520141601562501121008359375000000009556588121208364574226
1428211099074564513775851929601904249187532800039495538704384000050140820620800000004332166901637120000027025902314611200000012606834898944000000004487491524087000000000123113622060000000000002606930947120500000000042306317398800000000000517199626764843750000004614070407187500000000028375524096679687500000107616802500000000000001898270014801025390625225650314426128665359041
Technical details
Node id
n-fbc65803a7c7ccb7 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-23cc3a53c9ba374d

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:        36,         720,          2025;
  1:       216,        6480,         36450,          54000;
  2:      1296,       51840,        437400,        1296000,         1265625;
  3:      7776,      388800,       4374000,       19440000,        37968750,        27337500;
  4:     46656,     2799360,      39366000,      233280000,       683437500,       984150000,        558140625;
  5:    279936,    19595520,     330674400,     2449440000,      9568125000,     20667150000,      23441906250,      10935000000;
  6:   1679616,   134369280,    2645395200,    23514624000,    114817500000,    330674400000,     562605750000,     524880000000,     207594140625;
  7:  10077696,   906992640,   20407334400,   211631616000,   1240029000000,   4464104400000,   10126903500000,   14171760000000,   11210083593750,    3844335937500;
  8:  60466176,  6046617600,  153055008000,  1813985280000,  12400290000000,  53569252800000,  151903552500000,  283435200000000,  336302507812500,  230660156250000,   69774697265625;
  9: 362797056, 39907676160, 1122403392000, 14965378560000, 116917020000000, 589261780800000, 2005126893000000, 4676680800000000, 7398655171875000, 7611785156250000, 4605130019531250, 1245564843750000;
%F Formula
The row polynomials satisfy (4+(30*n+50)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (24+(180*n+540)*x+(150*n^2+1350*n+2550)*x^2+(375*n^2+2250*n+3375)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 720*x + 2025*x^2.
%t Mathematica
P[0, x_] := 36 + 720*x + 2025*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((24 + (180*n + 540)*x + (150*n^2 + 1350*n + 2550)*x^2 + (375*n^2 + 2250*n + 3375)*x^3)*P[n - 1, x])/(4 + (30*n + 50)*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] = 36 + 720*x + 2025*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((24 + (180*n + 540)*x + (150*n^2 + 1350*n + 2550)*x^2 + (375*n^2 + 2250*n + 3375)*x^3)*v[i - 1])/(4 + (30*n + 50)*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.