Real-rooted atlas

A038336 → x ↦ x/(x+1) → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(6 + 22\,x + 11\,n x\right)\,P_{n}(x) = \left(72 + 528\,x + 726\,x^{2} + 132\,n x + 242\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 1056\,x + 1452\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0144105614522652
11728190085227242592115600
2207363041281254528204441611712804795088
3248832456192025090560613324807027680030921792192432384
42985984656916484516300801471979520252996480022263690247936593287542280384
53583180891968307275873853443091156992070839014400935074990086666738355219954863104290423230208
642998169612612796416121398165504593502142464170013634560029922399682563200034410496191566685798449388286182411029903530240
7515978035217027275161618730002677761068303856435236722945064960807904791429121152012387778561034460103311365333934907699212072692177920414304185935872
8619173642242270303354880280950040166401831378039603207344589012992001938971499429888345603716333568041378404132454403200360944619520144872306135040029215915070566415422116162681856
974300837068829968004284416412060058910720302177376534528013849796424499200426573729874575369123938111206195213654873363709952014081588156325888095615722049126400385650078931476487011819616935936569768260120498176
10891610044825639230841972326459336648483143684834838024552448024929633564098560087752310145626931221897451466894868483932603528748466176506937173627731968045895546583580672002776680568306630656100970202483877478416711503420363980820916275404133650432
111069932053790725100009456402432841501560306401287542347318301818884321136484443750400171117004783972515844880003469765142118410224769174746012057615816439817185237401617899263167596462080014438738955194479411278756757937424433152260699453357678100483959340810363158528763654618781792337920
121283918464548864659078145135083521178102184428961792115192213588609597447259509293865500672031941840893008202956810248007286506798448642453944601939042893824442860314881186647244860141524243124112588806064270361181681352704441037844449576825651221898754082044960440326651692561410106327049332731910141730816027749299709983379079168
131540702157458636884738618660225024016312184092093317120172788320382914396160118791970263253647360057495313607414765322242049601457301359689728055213753543628465111040113878366683733709291520180424572729372337766400218313733002540528697344198467030002309571543040131392524492269762641920598652330526909569433601679891743825511546880021900810882465928314881004167640967117834747904
1418488425889503641610846543188508803072223709953262994063360255195057796304338944019006715242120583577600100355456478396681289728393523479801861060427776117789340893074058903552027330808004096090229964804948788280577069835878400698603945608129691831500876211339520886875472527366306841175628948606812160383137491537222124437504016126960740724910850048004204955689433458236456965119314543776410743603236205209121023226547339264
Technical details
Node id
n-c41436b6f6c89a56 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-efd843846d9eae45

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          144,           1056,            1452;
  1:         1728,          19008,           52272,            42592;
  2:        20736,         304128,         1254528,          2044416,           1171280;
  3:       248832,        4561920,        25090560,         61332480,          70276800,          30921792;
  4:      2985984,       65691648,       451630080,       1471979520,        2529964800,        2226369024,         793659328;
  5:     35831808,      919683072,      7587385344,      30911569920,       70839014400,       93507499008,       66667383552,        19954863104;
  6:    429981696,    12612796416,    121398165504,     593502142464,     1700136345600,     2992239968256,     3200034410496,      1915666857984,       493882861824;
  7:   5159780352,   170272751616,   1873000267776,   10683038564352,    36722945064960,    80790479142912,   115201238777856,    103446010331136,     53339349076992,    12072692177920;
  8:  61917364224,  2270303354880,  28095004016640,  183137803960320,   734458901299200,  1938971499429888,  3456037163335680,   4137840413245440,   3200360944619520,  1448723061350400,   292159150705664;
  9: 743008370688, 29968004284416, 412060058910720, 3021773765345280, 13849796424499200, 42657372987457536, 91239381112061952, 136548733637099520, 140815881563258880, 95615722049126400, 38565007893147648, 7011819616935936;
%F Formula
The row polynomials satisfy (6+(11*n+22)*x)*P(n,x) = (72+(132*n+528)*x+(242*n+726)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 1056*x + 1452*x^2.
%t Mathematica
P[0, x_] := 144 + 1056*x + 1452*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((72 + (132*n + 528)*x + (242*n + 726)*x^2)*P[n - 1, x])/(6 + (11*n + 22)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 144 + 1056*x + 1452*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((72 + (132*n + 528)*x + (242*n + 726)*x^2)*v[i - 1])/(6 + (11*n + 22)*x)); 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.