Real-rooted atlas

A038322 → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-5 + 24\,x + 24\,n\right)\,P_{n}(x) = \left(361 + 912\,x + 576\,x^{2} + 456\,n + 576\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1432467
2127320645763913
332851916567430413824212635
478878531536964399488237772833177611051473
51811461994654200189221760175979520713318407962624557264563
64036041372608505136681510240090826444806335262720205435699219110297627490578841
78797579747678054950162191144314243816457344003814710681602128648273925752199577645864714241333807603339
8188606936929168913531142465093275215361402332361113618318995251200146484890173446811674476544157774616985611007531417663877373610145
93991137314635407390983766641824266136337924686715815505927572594750013447913805948518405273456046243842101888124190724259914658611226418075402243027243871898659
1083576113724761957872955512400488869180519968014594129090703360281202948930355203634845480006451231655223794073600180804207299788806305664372572160113597724229632063403380965376142233709178126569
111735091750957683220640940233369041264392301276368004302048788575718409632125199864217601484751570352275456159933201120283852811938541545193472005966538840893030401849661549287833602998979919662284815216811431690246634725070178264443
1235762509481753233499706424275812704317722953936051417612138166092253132800309747512777451724805548104115121789337671268075376909221888658010884609167851524297874956269649920019092924290857697280532702526194896076878518746987521638436520347436056576307596227764974716017
137326072382108992911115790295830700869677954202187026781824330431872093493474304946776955195744358400193282447973129876275228850141398633304555523176519931085096747008256624244997575462092814899299848401453056005956992378747601551361510938074298250690562041487421675562598487648833846535782414185668933028214622355
1414928809129101255945246156032038862161776187452769699557746092887308706449469995642882775627725585345184153663623411389154020884480108238170864952730714112138480678713439861866496133413837105574063374336958063847990948391813125006164749062888226816018195903993265401102336423062660803510193356852764597975614541004821035720123168587776651222103102988207607097
Technical details
Node id
n-562222b4bedf067e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2fec102d566c1c30

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:             1;
  1:            43,             24;
  2:          1273,           2064,             576;
  3:         32851,          91656,           74304,           13824;
  4:        788785,        3153696,         4399488,         2377728,          331776;
  5:      18114619,       94654200,       189221760,       175979520,        71331840,         7962624;
  6:     403604137,     2608505136,      6815102400,      9082644480,      6335262720,      2054356992,       191102976;
  7:    8797579747,    67805495016,    219114431424,    381645734400,    381471068160,    212864827392,     57521995776,      4586471424;
  8:  188606936929,  1689135311424,   6509327521536,  14023323611136,  18318995251200,  14648489017344,   6811674476544,   1577746169856,   110075314176;
  9: 3991137314635, 40739098376664, 182426613633792, 468671581550592, 757259475001344, 791380594851840, 527345604624384, 210188812419072, 42599146586112, 2641807540224;
%F Formula
The row polynomials satisfy (24*n-5+24*x)*P(n,x) = (456*n+361+(576*n+912)*x+576*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((456*n + 361 + (576*n + 912)*x + 576*x^2)*P[n - 1, x])/(24*n - 5 + 24*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((456*n + 361 + (576*n + 912)*x + 576*x^2)*v[i - 1])/(24*n - 5 + 24*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.