Real-rooted atlas

A038319 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(32 + 40\,x + 16\,n + 20\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 640 + 800\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
06408001440
115360384002400077760
2327680122880015360006400003732480
3655360032768000614400005120000016000000167961600
41258291207864320001966080000245760000015360000003840000007255941120
5234881024017616076800550502400009175040000086016000000430080000008960000000304749527040
6429496729603758096384001409286144000293601280000036700160000002752512000000114688000000020480000000012538266255360
77730941132807730941132800338228674560008455716864000013212057600000013212057600000082575360000000294912000000004608000000000507799783342080
8137438953472001546188226560007730941132800002254857830400000422785843200000052848230400000004404019200000000235929600000000073728000000000010240000000000020311991333683200
92418925581107203023656976384000170080704921600005669356830720000012401718067200000018602577100800000019377684480000000013841203200000000064880640000000000180224000000000002252800000000000804354856813854720
1042221246506598405805421394657280036283883716608000013606456393728000003401614098432000000595282467225600000074410308403200000006643777536000000000415236096000000000017301504000000000004325376000000000004915200000000000031589208922144112640
1173183493944770560109775240917155840075470478130544640003144603255439360000088441966559232000000176883933118464000000257955735797760000000276381145497600000000215922769920000000000119957094400000000000449839104000000000001022361600000000000010649600000000000001231979147963620392960
121261007895663738880204913783045357568001536853372840181760007043911292184166400002201222278807552000000495275012731699200000082545835455283200000001031822943191040000000096733400924160000000006717597286400000000000335879864320000000000011450449920000000000002385510400000000000002293760000000000000047762883890281898311680
132161727821137838080037830236869912166400030737067456803635200001536853372840181760000052829334691381248000000132073336728453120000000247637506365849600000000353767866236928000000000386933603696640000000000322444669747200000000000201527918592000000000000916035993600000000000002862612480000000000000055050240000000000000004915200000000000000001842282664339444649164800
1436893488147419103232069175290276410818560006052837899185946624000032786205287257210880000012294826982721454080000003381077420248399872000000704391129218416640000000011320571719581696000000000141507146494771200000000001375763924254720000000000010318229431910400000000000586263035904000000000000024427626496000000000000007046430720000000000000001258291200000000000000001048576000000000000000070743654310634674527928320
Technical details
Node id
n-cfe169c7cfe15942 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-6b726760bd05165a

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:             640,              800;
  1:           15360,            38400,             24000;
  2:          327680,          1228800,           1536000,            640000;
  3:         6553600,         32768000,          61440000,          51200000,           16000000;
  4:       125829120,        786432000,        1966080000,        2457600000,         1536000000,          384000000;
  5:      2348810240,      17616076800,       55050240000,       91750400000,        86016000000,        43008000000,         8960000000;
  6:     42949672960,     375809638400,     1409286144000,     2936012800000,      3670016000000,      2752512000000,      1146880000000,       204800000000;
  7:    773094113280,    7730941132800,    33822867456000,    84557168640000,    132120576000000,    132120576000000,     82575360000000,     29491200000000,     4608000000000;
  8:  13743895347200,  154618822656000,   773094113280000,  2254857830400000,   4227858432000000,   5284823040000000,   4404019200000000,   2359296000000000,   737280000000000,   102400000000000;
  9: 241892558110720, 3023656976384000, 17008070492160000, 56693568307200000, 124017180672000000, 186025771008000000, 193776844800000000, 138412032000000000, 64880640000000000, 18022400000000000, 2252800000000000;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (16*n+32+(20*n+40)*x)*P(n-1,x) for n >= 1, with P(0,x) = 640 + 800*x.
%t Mathematica
P[0, x_] := 640 + 800*x;
P[n_, x_] := P[n, x] = Expand[Together[((16*n + 32 + (20*n + 40)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 640 + 800*x; for(n = 1, N, my(i = n + 1); v[i] = ((16*n + 32 + (20*n + 40)*x)*v[i - 1])/(n + 1)); 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.