Real-rooted atlas

A038303 → basis A013611 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(11\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
144
28832120
3145210561922700
421296232328448102454000
52928204259202323205632051201012500
638652247027680511104018585603379202457618225000
7496037081082262729838752047703040130099201892352114688318937500
862358947215873186561731620352104946688038162432083263488100925445242885467500000
977169197162244922099228571735808207794442249445201920274769510449958092851904512235929692264062500
109431790764030867678864044898441984038095647744020779444224075561615360183179673602854748160259522560104857601537734375000
111141246682444414998793616067908893500806585104824320419052125184018285910917125541185126401151415091201570111488012687769604613734425372617187500
121369496018932854779840757312995997104678401086542296012807902125789184040229004017664146287287336963799669800960690849054720837392793606090129408201326592415188281250000
13163198275589492712137929845056142427585969011217263949814425601412504984816640821821082075136348651368153088108670556307456246978537062403991572316160435444252672287897026568724152326746809570312500
141933271880060136913910343301155219939862035661568265864827142154242416952974019584015820055829946368767033009936793627892109452247047606938941521921536755341721602235280497049622168071045121343519457283758096384108986923828125000
Technical details
Node id
n-f74c73498eaf58f5 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-432dc7cc45625337

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:          4;
  2:         88,          32;
  3:       1452,        1056,         192;
  4:      21296,       23232,        8448,        1024;
  5:     292820,      425920,      232320,       56320,       5120;
  6:    3865224,     7027680,     5111040,     1858560,     337920,      24576;
  7:   49603708,   108226272,    98387520,    47703040,   13009920,    1892352,    114688;
  8:  623589472,  1587318656,  1731620352,  1049466880,  381624320,   83263488,  10092544,   524288;
  9: 7716919716, 22449220992, 28571735808, 20779444224, 9445201920, 2747695104, 499580928, 51904512, 2359296;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (11*n+4*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((11*n + 4*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((11*n + 4*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.