Real-rooted atlas

A178821 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-n + 2\,n x + 2\,n^{2}\right)\,P_{n}(x) = \left(8 + 32\,x + 32\,x^{2} + 18\,n + 40\,n x + 8\,n x^{2} + 4\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1302050
23007202401260
31960840010080224022680
4100806272013440010752017920332640
544352362880112896016128009676801290244245696
6174720177408072576001505280016128000774144086016048988800
7633600768768039029760106444800165580800141926400567705605406720523480320
8215424030412800184504320624476160127733760015895756801135411200389283840324403205265596160
969555201120204807907328003198074880811819008013284311040137763225608434483200253034496018743296050438868480
1021525504389509120313657344014760345600447730483209092372889612398690304011021058048059041382400157443686401049624576464037590016
11642969601291530240116852736006273146880022140518400053727657984090923728896010627448832008265793536003936092160009446621184057252249604126816512000
1218636800041150054404132896768024928583680010037035008002833986355200573095018496083130266419208501959065600587789762560025190989824005496215961603053453312035655694663680
13526417920126730240001399101849609367899340804237859225600136503676108803211851202560055672087511040706607264563206423702405120039969703854080155726118912003114522378240159719096320300483033661440
141453916160379020902404562288640003357844439040168622188134406102517284864016380441133056033036183797760050104878759936056528581165056046250657316864026161987977216093435671347200172496624025608214124953602477874868715520
Technical details
Node id
n-de8788eb877ead40 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4be2d3b20c0d29e0

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:      30,        20;
  2:     300,       720,       240;
  3:    1960,      8400,     10080,       2240;
  4:   10080,     62720,    134400,     107520,      17920;
  5:   44352,    362880,   1128960,    1612800,     967680,      129024;
  6:  174720,   1774080,   7257600,   15052800,   16128000,     7741440,      860160;
  7:  633600,   7687680,  39029760,  106444800,  165580800,   141926400,    56770560,    5406720;
  8: 2154240,  30412800, 184504320,  624476160, 1277337600,  1589575680,  1135411200,  389283840,   32440320;
  9: 6955520, 112020480, 790732800, 3198074880, 8118190080, 13284311040, 13776322560, 8434483200, 2530344960, 187432960;
%F Formula
The row polynomials satisfy (2*n^2-n+2*n*x)*P(n,x) = (4*n^2+18*n+8+(8*n^2+40*n+32)*x+(8*n+32)*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[((4*n^2 + 18*n + 8 + (8*n^2 + 40*n + 32)*x + (8*n + 32)*x^2)*P[n - 1, x])/(2*n^2 - n + 2*n*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] = ((4*n^2 + 18*n + 8 + (8*n^2 + 40*n + 32)*x + (8*n + 32)*x^2)*v[i - 1])/(2*n^2 - n + 2*n*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.