Real-rooted atlas

A038334 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(9 + 4\,x + 4\,n\right)\,P_{n}(x) = \left(260 + 288\,x + 64\,x^{2} + 80\,n + 64\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1040 + 1152\,x + 256\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0104011522562448
12720049920276484096108864
267200017408001597440589824655364665600
316000000537600006963200042598400117964801048576194835456
43712000001536000000258048000022282240001022361600226492416167772167981535232
5844800000041574400000860160000009633792000062390272000229008998404227858432268435456322163785728
61894400000001081344000000266076160000036700160000003082813440000159719096320048855252992077309411328429496729612851722911744
741984000000002727936000000077856768000000127716556800000132120576000000887850270720003833258311680010050223472640139156940390468719476736507799783342080
8921600000000006717440000000002182348800000000415236096000000051086622720000004227858432000000236760072192000087617332838400020100446945280024739011624960109951162777619905751507009536
9200704000000000016220160000000000591134720000000001280311296000000001827038822400000001798249119744000001240171806720000005952824672256000019275813224448000393075406929920043540660459929617592186044416775105589293350912
1043417600000000000385351680000000000155713536000000000037832622080000000006145494220800000000701582907801600000057543971831808000003401614098432000000142867792134144000041121734878822400075470478130544640759982437118771228147497671065630009748476036907008
1193388800000000000090308608000000000004007657472000000000010796138496000000000019672963481600000000025565255958528000000024321540803788800000017098780201451520000088441966559232000000330183341821132800008553320854795059200142707813192302592013173028910058700845035996273704961156165046550474522624
1219988480000000000000209190912000000000000101145640960000000000029923842457600000000006045837557760000000000881348763975680000000095443622245171200000007782893057212416000000478765845640642560000022012222788075520000007396106856793374720001741767155885539328002663879179589648384022698142121947299847205759403792793644351249326690334146560
1342598400000000000000047972352000000000000002510290944000000000000080916512768000000000000179543054745600000000000290200202772480000000000352539505590272000000000327235276269158400000000233486791716372480000000127670892170838016000000528293346913812480000001613696041482190848000034835343117710786560004917930793088581632003891110078048108544011529215046068469761694900051192289077231616
1490439680000000000000001090519040000000000000006140461056000000000000002142114938880000000000000517865681715200000000000091926044029747200000000001238187531829248000000000012892873347301376000000000104715288406130688000000006641402075487928320000000326837483957345320960000012294826982721454080000003442551555162007142400006859882952410739507200089927877359334064128006640827866535438581761844674407370955161664553584558454140506734592
Technical details
Node id
n-80b2fb0627fa560a · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-374a4446546c6a5d

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:             1040,              1152,               256;
  1:            27200,             49920,             27648,               4096;
  2:           672000,           1740800,           1597440,             589824,              65536;
  3:         16000000,          53760000,          69632000,           42598400,           11796480,            1048576;
  4:        371200000,        1536000000,        2580480000,         2228224000,         1022361600,          226492416,           16777216;
  5:       8448000000,       41574400000,       86016000000,        96337920000,        62390272000,        22900899840,         4227858432,         268435456;
  6:     189440000000,     1081344000000,     2660761600000,      3670016000000,      3082813440000,      1597190963200,       488552529920,       77309411328,        4294967296;
  7:    4198400000000,    27279360000000,    77856768000000,    127716556800000,    132120576000000,     88785027072000,     38332583116800,    10050223472640,     1391569403904,      68719476736;
  8:   92160000000000,   671744000000000,  2182348800000000,   4152360960000000,   5108662272000000,   4227858432000000,   2367600721920000,   876173328384000,   201004469452800,   24739011624960,   1099511627776;
  9: 2007040000000000, 16220160000000000, 59113472000000000, 128031129600000000, 182703882240000000, 179824911974400000, 124017180672000000, 59528246722560000, 19275813224448000, 3930754069299200, 435406604599296, 17592186044416;
%F Formula
The row polynomials satisfy (4*n+9+4*x)*P(n,x) = (80*n+260+(64*n+288)*x+64*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1040 + 1152*x + 256*x^2.
%t Mathematica
P[0, x_] := 1040 + 1152*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((80*n + 260 + (64*n + 288)*x + 64*x^2)*P[n - 1, x])/(4*n + 9 + 4*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1040 + 1152*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((80*n + 260 + (64*n + 288)*x + 64*x^2)*v[i - 1])/(4*n + 9 + 4*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

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.