Real-rooted atlas

A136572 → basis A013611 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15813
250160128338
375036005760307213182
4150009600023040024576098304685464
5375000300000096000001536000012288000393216044555160
61125000010800000043200000092160000011059200007077888001887436803475302480
73937500004410000000211680000005644800000090316800000867041280004624220160010569646080316252525680
81575000000020160000000011289600000003612672000000722534400000092484403200007398752256000338228674560067645734912032890262670720
97087500000001020600000000065318400000000243855360000000585252864000000936404582400000998831554560000684913065984000273965226393600487049291366403848160732474240
1035437500000000567000000000000408240000000000017418240000000000487710720000000009364045824000000012485394432000000011415217766400000068491306598400000243524645683200003896394330931200500260895221651200
111949062500000000343035000000000002744280000000000001317254400000000000421521408000000000094420795392000000001510732726272000000017265516871680000000138124134973440000007366620531916800000235731857021337600034288270112194560071537308016696121600
1211694375000000000022453200000000000001975881600000000000010538035200000000000037936926720000000000097118532403200000000018128792715264000000002486234429521920000000248623442952192000000017679889276600320000008486346852768153600002468755448078008320003291673930770677760011159820050604594969600
13760134375000000000015810795000000000000015178363200000000000008904639744000000000000356185589760000000000001025814498508800000000002188404263485440000000003501446821576704000000004201736185892044800000003734876609681817600000002390321030196363264000001043049176812958515200002781464471501222707200034233408880015048704001886009588552176549862400
1453209406250000000000011918907000000000000000123956632800000000000000793322449920000000000000349061877964800000000000011169980094873600000000000268079522276966400000000004902025550207385600000000068628357702903398400000000732035815497636249600000005856286523981089996800000034072939775889978163200000136291759103559912652800003354874070241474772992000383414179456168545484800343253745116496132074956800
Technical details
Node id
n-773d2d719c60037e · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-4b28493176e02e0b

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:            5,              8;
  2:           50,            160,            128;
  3:          750,           3600,           5760,            3072;
  4:        15000,          96000,         230400,          245760,           98304;
  5:       375000,        3000000,        9600000,        15360000,        12288000,         3932160;
  6:     11250000,      108000000,      432000000,       921600000,      1105920000,       707788800,       188743680;
  7:    393750000,     4410000000,    21168000000,     56448000000,     90316800000,     86704128000,     46242201600,     10569646080;
  8:  15750000000,   201600000000,  1128960000000,   3612672000000,   7225344000000,   9248440320000,   7398752256000,   3382286745600,    676457349120;
  9: 708750000000, 10206000000000, 65318400000000, 243855360000000, 585252864000000, 936404582400000, 998831554560000, 684913065984000, 273965226393600, 48704929136640;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+8*n*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[(5*n + 8*n*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] = (5*n + 8*n*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.