Real-rooted atlas

A038331 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(144 + 300\,x + 100\,x^{2} + 180\,n x + 100\,n x^{2} + 25\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1728 + 6480\,x + 5100\,x^{2} + 1125\,x^{3} + 2160\,n x + 2700\,n x^{2} + 750\,n x^{3} + 300\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 480\,x + 225\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0144480225849
1172886408100200020468
2207361382401944009600015625465001
324883220736003888000288000093750011250010140432
42985984298598406998400069120000337500008100000765625214565449
535831808418037760117573120014515200009450000003402000006431250050000004435633268
642998169657330892801881169920027869184000226800000001088640000030870000004800000003164062590008994801
75159780352773967052802902376448005016453120004898880000002939328000001111320000002592000000034171875001953125001798924742432
861917364224103195607040043535646720008599633920000979776000000070543872000003333960000000103680000000020503125000023437500000118164062535499629617249
974300837068813621820129280638522818560001418939596800001847577600000001551965184000008801654400000034214400000000902137500000015468750000001559765625007031250000693027550248468
1089161004482561783220089651209194728587264002270303354880000332563968000000031926140928000002112397056000000985374720000000324769500000000742500000000001123031250000010125000000004125976562513404343444085401
111069932053790722318186116546560130397969055744003541673233612800057644421120000000622559748096000004707627724800000025619742720000000101328084000000002895750000000000583976250000000789750000000006436523437500239257812500257176309892478032
121283918464548864299580975061401601825571566780416005409100938608640009684262748160000001162111529779200000988601822208000000614873825280000000283718635200000000972972000000000002452700250000000044226000000000005406679687500004019531250000013732910156254899270392865060249
1315407021574586368385175539364659200252771447708057600081136514079129600001584697540608000000020918007536025600000197720364441600000001383466106880000000072956220480000000002918916000000000000882972090000000000199017000000000000324400781250000003617578125000000247192382812500781250000000092746468700131194068
14184884258895036416493024690386763776034665798542819328000119832390024560640000253551606497280000000365114313356083200000379623099727872000000295139436134400000000175094929152000000000800616960000000000002825510688000000000076422528000000000001557123750000000000231525000000000000237304687500000001500000000000000440979003906251745909683594428232801
Technical details
Node id
n-731c6b3b4289d4ac · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-399776af7ffddb1e

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:          144,            480,            225;
  1:         1728,           8640,           8100,            2000;
  2:        20736,         138240,         194400,           96000,           15625;
  3:       248832,        2073600,        3888000,         2880000,          937500,          112500;
  4:      2985984,       29859840,       69984000,        69120000,        33750000,         8100000,         765625;
  5:     35831808,      418037760,     1175731200,      1451520000,       945000000,       340200000,       64312500,        5000000;
  6:    429981696,     5733089280,    18811699200,     27869184000,     22680000000,     10886400000,     3087000000,      480000000,      31640625;
  7:   5159780352,    77396705280,   290237644800,    501645312000,    489888000000,    293932800000,   111132000000,    25920000000,    3417187500,     195312500;
  8:  61917364224,  1031956070400,  4353564672000,   8599633920000,   9797760000000,   7054387200000,  3333960000000,  1036800000000,  205031250000,   23437500000,   1181640625;
  9: 743008370688, 13621820129280, 63852281856000, 141893959680000, 184757760000000, 155196518400000, 88016544000000, 34214400000000, 9021375000000, 1546875000000, 155976562500, 7031250000;
%F Formula
The row polynomials satisfy (144+(180*n+300)*x+(25*n^2+100*n+100)*x^2)*P(n,x) = (1728+(2160*n+6480)*x+(300*n^2+2700*n+5100)*x^2+(125*n^2+750*n+1125)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 480*x + 225*x^2.
%t Mathematica
P[0, x_] := 144 + 480*x + 225*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1728 + (2160*n + 6480)*x + (300*n^2 + 2700*n + 5100)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*P[n - 1, x])/(144 + (180*n + 300)*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] = 144 + 480*x + 225*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1728 + (2160*n + 6480)*x + (300*n^2 + 2700*n + 5100)*x^2 + (125*n^2 + 750*n + 1125)*x^3)*v[i - 1])/(144 + (180*n + 300)*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.