Real-rooted atlas

A038321 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(16 + 7\,x + 7\,n\right)\,P_{n}(x) = \left(414 + 448\,x + 98\,x^{2} + 126\,n + 98\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 828 + 896\,x + 196\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
08288961961920
1194403477618816274475776
24315681088640973728351232384162883584
392378883020976038102400227203206146560537824106954752
41927359367759825921268809920106686720047712672010326220875295363892314112
539454179841888812172838023147008414477907202613824640093516837121686616064105413504139586437120
6795886041604418868142081057734816768141953082163211605381401605854967193601745647626242698585702414757890564947802324992
71586874322944100281641241602783886929510444424862304256447152208814082924556113203212295431106560314216572723242502724812820661046784173722837188608

All 15 rows

Technical details
Node id
n-37bb950516e82028 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7e8a57fb8b78011b

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:             828,              896,               196;
  1:           19440,            34776,             18816,              2744;
  2:          431568,          1088640,            973728,            351232,             38416;
  3:         9237888,         30209760,          38102400,          22720320,           6146560,            537824;
  4:       192735936,        775982592,        1268809920,        1066867200,         477126720,         103262208,           7529536;
  5:      3945417984,      18888121728,       38023147008,       41447790720,       26138246400,        9351683712,        1686616064,         105413504;
  6:     79588604160,     441886814208,     1057734816768,     1419530821632,     1160538140160,      585496719360,      174564762624,       26985857024,       1475789056;
  7:   1586874322944,   10028164124160,    27838869295104,    44424862304256,    44715220881408,    29245561132032,    12295431106560,     3142165727232,     425027248128,     20661046784;
  8:  31340767878144,  222162405212160,   701971488691200,  1299147233771520,  1554870180648960,  1252026184679424,   682396426414080,   245908622131200,   54987900226560,   6611534970880,    289254654976;
  9: 614120362979328, 4826478253234176, 17106505201336320, 36034536419481600, 50017168500203520, 47890001563987968, 32135338740105216, 15012721381109760, 4733740976025600, 940904070543360, 101817638551552, 4049565169664;
%F Formula
The row polynomials satisfy (7*n+16+7*x)*P(n,x) = (126*n+414+(98*n+448)*x+98*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 828 + 896*x + 196*x^2.
%t Mathematica
P[0, x_] := 828 + 896*x + 196*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((126*n + 414 + (98*n + 448)*x + 98*x^2)*P[n - 1, x])/(7*n + 16 + 7*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 828 + 896*x + 196*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((126*n + 414 + (98*n + 448)*x + 98*x^2)*v[i - 1])/(7*n + 16 + 7*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.