Real-rooted atlas

A038303 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(11\,n + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
222224
3363663432
45324145213246912
5732052662036302205103680
696630643923079860726033061492992
7124009276764142153730518634012705462720901888
815589736899207416270565684099480372680203286168286654464
9192922992914030763124464333728116970492238306708243049279293869835264
10235794769101929229929070153815601488111240202924260184476601118040435609901051597803520
1128531167061125937424601010610764609525723065720409230591044643337233820710175692059895121011681091006464
12342374004733234237400473321556245476060424430584380771691971609821534184892866744579783602635380798601452128916100448256
1340799568897373445086206153162225431030765867437303962601379399399235200639912616212799907321658181096942148353806660103818171613115909305827328
1448331797001503457119396456322231156034430721210385344810240423603056386910386231831785846815979610442559981464290181691814655641053293241321322002141497904875307008
Technical details
Node id
n-5fa66095534ffc89 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b6561930bb158c66

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:          1;
  2:         22,          2;
  3:        363,         66,         3;
  4:       5324,       1452,       132,        4;
  5:      73205,      26620,      3630,      220,       5;
  6:     966306,     439230,     79860,     7260,     330,      6;
  7:   12400927,    6764142,   1537305,   186340,   12705,    462,     7;
  8:  155897368,   99207416,  27056568,  4099480,  372680,  20328,   616,   8;
  9: 1929229929, 1403076312, 446433372, 81169704, 9223830, 670824, 30492, 792, 9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (11*n+n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 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] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((11*n + 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.