Real-rooted atlas

A109692 → basis A007318 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-1 - x + 4\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
22116340
323128111315640
4346558323662101610514080
5658351419931220665226211139945394240
615142053990024436944125447528310791442641039513404160
74088353512741531316984521912551103755520909146991552156097135135536166400
812673895854563127728717643148464385204323603813734128848744028732633236530640202702524663654400
944358635475181255093425328748273316347347550348235588328034106361893878319607081006163120044691966755344594251282510028800
1017299867835257911762717600162650294343991979277165657615787548249938862429209388832673435995828476151356161440859842811449128592065472907574385581670400
117438943169157537653551931082586554328274875711926547993551471094512779535430702383073285882321605908753674105061666422710239956151720193648930964461332470354545137493105754760677226905600
123496303289504025194081263365150004934085123334055475962271072913320788731610233335915818579837375134431270288396997964123348342232018723544212240807223723398904286104995515188462188293035751560316234143225333247405883392000
131783114677647052751077222025399865625300158657131324325451075971055520931705921587989012846141493930474264465831930494296675906797641410833755308244572489125679861215173125498650138771879231621000683097182911732794485010223453835093475790585358062525326802847137792000
14980713072705879012564061621026639651800194172256108024750845361960678230822692208463592461245613053345431544636548803053352301079859768893161633159930457566901805088649944237856894379592010881911388526301646623790575038752237855220055410379369090674638123639564680754944582002134580466768752076797833465298944000
Technical details
Node id
n-f6425a025280c66f · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a46d1ee90c8acbaf

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:           3,            1;
  2:          21,           16,            3;
  3:         231,          281,          113,           15;
  4:        3465,         5832,         3662,         1016,          105;
  5:       65835,       141993,       122066,        52262,        11139,          945;
  6:     1514205,      3990024,      4369441,      2544752,       831079,       144264,       10395;
  7:    40883535,    127415313,    169845219,    125511037,     55520909,     14699155,     2156097,     135135;
  8:  1267389585,   4563127728,   7176431484,   6438520432,   3603813734,   1288487440,   287326332,   36530640,   2027025;
  9: 44358635475, 181255093425, 328748273316, 347347550348, 235588328034, 106361893878, 31960708100, 6163120044, 691966755, 34459425;
%F Formula
The row polynomials satisfy P(n,x) = (4*n-1+(2*n-1)*x)*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[(4*n - 1 + (2*n - 1)*x)*P[n - 1, 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 - 1 + (2*n - 1)*x)*v[i - 1]); 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.