Real-rooted atlas

A038303 → basis A007318 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(12 + 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112315
2144729225
317281296324273375
42073620736777612968150625
5248832311040155520388804860243759375
62985984447897627993609331201749601749672911390625
7358318086270566447029248195955204898880734832612362187170859375
842998169685996339275246796837623398411757312023514624293932820995265612562890625
95159780352116095057921160950579267722117122539579392634894848105815808113374087085881968338443359375
10619173642241547934105601741425868801160950579205079158784015237476352317447424045349632042515280236196059049576650390625
117430083706882043273019392255409127424019155684556809577842278403352244797448380611993614965378560187067232015588936077944681771478649755859375
128916100448256267483013447683677891434905630649095290880172401161011206896046440448201134687846443100290252867344203520748268928056120169625509168531441129746337890625
13106993205379072347727917481984521591876222976478125886537728298828679086080134472905588736448243018629121120607546572821011391498242918248819202918248819219897151048290479615943231946195068359375
1412839184645488644493714625921024730228626712166473022862671216645020321808646144251016090432307294131033912115226894581117747258831896195072980531603251212256645040641114240458246964002864267846264478296929192926025390625
Technical details
Node id
n-be5fbaa0cf892ed1 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-03d7b8417365c509

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:         12,           3;
  2:        144,          72,           9;
  3:       1728,        1296,         324,         27;
  4:      20736,       20736,        7776,       1296,         81;
  5:     248832,      311040,      155520,      38880,       4860,       243;
  6:    2985984,     4478976,     2799360,     933120,     174960,     17496,       729;
  7:   35831808,    62705664,    47029248,   19595520,    4898880,    734832,     61236,     2187;
  8:  429981696,   859963392,   752467968,  376233984,  117573120,  23514624,   2939328,   209952,   6561;
  9: 5159780352, 11609505792, 11609505792, 6772211712, 2539579392, 634894848, 105815808, 11337408, 708588, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (12+3*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[(12 + 3*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] = (12 + 3*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.