Real-rooted atlas

A196347 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(6\,n + 4\,n x\right)\,P_{n-1}(x) + \left(4\,n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
27212832232
31296470428803849264
431104208896236544737286144556416
5933120110668802073600011366400215040012288046375680
633592320686407680200742912017252352005566464007077888029491205083038720
714108774404898912256021594746880027358765056013403013120028467855360260112384082575360705116805120
86772211712039642778828802575376842752046306940682240323441079091201021027811328015352410931201056964608002642411520120290675097600
9365699432448035922300567552033876888256512008428860053913600807118323056640035535929632358407820851072204808761179635712047087773286409512681472024678705881088000
102194196594688003606858034053120048864535581818880016527023627304960002112814712881152000124678149966397440037676925069557760061071415050240000529380723916800022830435532800038050725888005980598513506713600
1114481697524940800397719830247505920076838402330984448000348735463476638515200584020064477380608000449543356270667366400179192238714755481600394185562593951744004899953688182784000338404130684928000119707583643648001674231939072001686990257530247577600
121042682221795737600477958917778204262400130997571101241901056007901510342284117278720017092059926890712924160016829049025136248750080086087287038304006963200246839984121414942720004126469582262317875200405753458897977344000228512568900059136006750503178338304008036313307545600547131994488361923379200
13813292133000675328006218887878669993246720024090096577670692208640001917072157067129231769600052992076768895179948032000657700568669436914958336004232514743800941206568960015386866224158348830310400332204389106894273249280043611780683593615933440034764231482918358220800162704806487229726720040744108469256192000417888291992371200201910742873894424674304000
146831653917205672755200871099746608279432724480047564847013440076350750720049659263766441215404277760001739512005277730779378483200026907856998625633781573222400214596851085127997913563136009694356579227901118906368000262899185860154506818355200044247172831939125470822400046810762257811847616921600308573060845403614740480012191723763560632811520026209953673761521664002340174435157278720084028796056464114330265190400
Technical details
Node id
n-4c6d389a1e69ba8f · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-5162632e955115eb

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             1;
  1:             6,               4;
  2:            72,             128,               32;
  3:          1296,            4704,             2880,              384;
  4:         31104,          208896,           236544,            73728,             6144;
  5:        933120,        11066880,         20736000,         11366400,          2150400,           122880;
  6:      33592320,       686407680,       2007429120,       1725235200,        556646400,         70778880,         2949120;
  7:    1410877440,     48989122560,     215947468800,     273587650560,     134030131200,      28467855360,      2601123840,       82575360;
  8:   67722117120,   3964277882880,   25753768427520,   46306940682240,   32344107909120,   10210278113280,   1535241093120,   105696460800,    2642411520;
  9: 3656994324480, 359223005675520, 3387688825651200, 8428860053913600, 8071183230566400, 3553592963235840, 782085107220480, 87611796357120, 4708777328640, 95126814720;
%F Formula
The row polynomials satisfy P(n,x) = (6*n+4*n*x)*P(n-1,x) + 4*n*x*d/dx 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[(6*n + 4*n*x)*P[n - 1, x] + 4*n*x*D[P[n - 1, x], 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] = (6*n + 4*n*x)*v[i - 1] + 4*n*x*deriv(v[i - 1], 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

  • Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.