Real-rooted atlas

A038271 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(15 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1151227
2225360144729
3337581006480172819683
45062516200019440010368020736531441
5759375303750048600003888000155520024883214348907
6113906255467500010935000011664000069984000223948802985984387420489
717085937595681250022963500003061800000244944000011757312003135283203583180810460353203
825628906251640250000045927000000734832000007348320000047029248000188116992004299816960429981696282429536481
9384433593752767921875008857350000001653372000000198404640000015872371200008465264640002902376448005804752896051597803527625597484987
105766503906254613203125000166075312500003542940000000049601160000000476171136000003174474240000014511882240000435356467200077396705280061917364224205891132094649
1186497558593757611785156250030447140625000073073137500000011691702000000001309470624000000104757649920000059861514240000023944605696000063852281856000102163650969607430083706885559060566555523
121297463378906251245564843750000548048531250000014614627500000000263063295000000003367210176000000031427294976000000215501451264000001077507256320000038311369113600009194728587264001337415067238408916100448256150094635296999121
1319461950683593752024042871093750097154057812500000284985236250000000569970472500000000820757480400000000875474645760000000700379716608000000420227829964800000186767924428800000597657358172160001303979690557440017386395874099201069932053790724052555153018976267
142919292602539062532696077148437500017001960117187500005440627237500000000119693799225000000001915100787600000000022981209451200000000210113914982400000001470797404876800000078442528260096000003137701130403840000912785783390208000182557156678041600224685731296051201283918464548864109418989131512359209
Technical details
Node id
n-2b363192a9cb3eca · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-6b726760bd05165a

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:          15,           12;
  2:         225,          360,          144;
  3:        3375,         8100,         6480,          1728;
  4:       50625,       162000,       194400,        103680,         20736;
  5:      759375,      3037500,      4860000,       3888000,       1555200,        248832;
  6:    11390625,     54675000,    109350000,     116640000,      69984000,      22394880,      2985984;
  7:   170859375,    956812500,   2296350000,    3061800000,    2449440000,    1175731200,    313528320,     35831808;
  8:  2562890625,  16402500000,  45927000000,   73483200000,   73483200000,   47029248000,  18811699200,   4299816960,   429981696;
  9: 38443359375, 276792187500, 885735000000, 1653372000000, 1984046400000, 1587237120000, 846526464000, 290237644800, 58047528960, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (15+12*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[(15 + 12*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] = (15 + 12*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.