Real-rooted atlas

A038326 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(23\,n + 24\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 24\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
12424
2110411522256
3380887948841472159048
411680323656448381542413271049967008
53358092014016384021938688015261696039813120585561720
6926833392483565248010091796480105305702405494210560114661785633025681008
724870029352155708009856406194808320565140602880442283950080184605474816321052999681810908175272
8653726485824477504563558414947968946176259964677324802712674893824016983703683072590737519411288060251340897271639128896
916915172820696141204920937984515704928643072107625376412467214038092575539201171875554131968611413332590592182284720275456237762678620165143237918940376
1043227663875112040596414769670401694459051255808041256394291445760645752258474803206738284436258816046875022165278720209627428316774405468541608263680634033809653760268591313544664080
1110936598960403336114121032630295680535872674959649280149112396510511104027229220232354201603409571924746960896296484515195387904017678579788047974406917705134453555201604105538424012801673849257485926413886170910259132936
122744092102792109763149740500596160768164334286987625779205144377679612633088010736092548756799488015684030853836020121616365945238785412300812198220053753102336063642887236972707840221366564302513766404619823950661156864438244169232678912711981853944195543264
1368373628227903401528561567360711382451249135951809300107980817090765846713081036804012614590097853808640669932175042424288051281556960439947304632327294535592144355196928475730582096370991104022062867575483872051206906636806238429511681310350066005709946881139434840004965171236251742729991956411192
1416935621761065304068822973539084575542910721438343316599512251801655032266026416120938496143562433112389880709120269647700454575775940608375162018023757601308672391473410111747062235136306370494870062918270976177606083982645170012160741312350536258100920322109663606269193014476836689801848159878512642945000817243602288641834895901256515947581872
Technical details
Node id
n-55fdb7333755864e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-723ac5bf98d71a32

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:             24;
  2:           1104,            1152;
  3:          38088,           79488,           41472;
  4:        1168032,         3656448,         3815424,          1327104;
  5:       33580920,       140163840,       219386880,        152616960,         39813120;
  6:      926833392,      4835652480,     10091796480,      10530570240,       5494210560,       1146617856;
  7:    24870029352,    155708009856,    406194808320,     565140602880,     442283950080,     184605474816,     32105299968;
  8:   653726485824,   4775045635584,  14947968946176,   25996467732480,   27126748938240,   16983703683072,   5907375194112,    880602513408;
  9: 16915172820696, 141204920937984, 515704928643072, 1076253764124672, 1403809257553920, 1171875554131968, 611413332590592, 182284720275456, 23776267862016;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (23*n+24*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 24.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 24;
P[n_, x_] := P[n, x] = Expand[Together[((23*n + 24*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] = 24; for(n = 2, N, my(i = n + 1); v[i] = ((23*n + 24*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.