Real-rooted atlas

A038221 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6\,n + 12\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 12\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11212
2144288432
312965184518411664
4103686220812441682944279936
5777606220801866240248832012441606298560
6559872559872022394880447897604478976017915904136048896
73919104470292482351462406270566409405849607524679682508226562857026816
82687385637623398422574039047524679680150493593601805923123212039487488343985356858773123072
918139852829023764482031663513681266540544203166351360325066162176325066162176185752092672464380231681190155742208
101209323520217678233601741425868808126654054402437996216320487599243264065013232435205572562780160278628139008061917364224023803114844160
11798153523215963070464014366763417607662273822720268179583795206436310011084810727183351808012259638116352091947285872640408654603878408173092077568471301673914368
12522427760641149341073408114934107340806896046440448027584185761792077235720133017615447144026603522206734860943360220673486094336014711565739622405884626295848961069932053790729254651051409408
1333957804441681498730659848964860372582459765735817216026894581117747208606265957679104200812872345845763442506383071641643031329788395520382500709230182402295004255381094483454700195676161390911669927936180465695502483456
1421941965946885704911146188868458933754265650203218086461442510160904323072090365792555630592240975446815014912481950893630029824722926340445044736803251489383383040642601191506706432350509740821839872116836580273946624179748585036840963498258097432756224
Technical details
Node id
n-66bf587f689cc703 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-11aec07ce60b6d61

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:         0;
  1:        12;
  2:       144,        288;
  3:      1296,       5184,        5184;
  4:     10368,      62208,      124416,       82944;
  5:     77760,     622080,     1866240,     2488320,      1244160;
  6:    559872,    5598720,    22394880,    44789760,     44789760,     17915904;
  7:   3919104,   47029248,   235146240,   627056640,    940584960,    752467968,    250822656;
  8:  26873856,  376233984,  2257403904,  7524679680,  15049359360,  18059231232,  12039487488,   3439853568;
  9: 181398528, 2902376448, 20316635136, 81266540544, 203166351360, 325066162176, 325066162176, 185752092672, 46438023168;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (6*n+12*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 12.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 12;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 12*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 12; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 12*n*x)*v[i - 1])/(n - 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.