Real-rooted atlas

A038255 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4 + 7\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14711
2165649121
3643365883431331
4256179247045488240114641
51024896031360548804802016807161051
64096430081881604390405762404033681176491771561
7163842007041053696307328053782405647152329417282354319487171
865536917504561971219668992430259206023628852706752263533765764801214358881
9262144412876828901376118013952309786624542126592632481024474360768207532836403536072357947691
1010485761835008014450688067436544020652441604337012736632481024063248102404150656720161414428028247524925937424601
114194304807403527064780803709009920129815347203180476006455658330112695729126406087629856035511174160124289109561977326743285311670611
12167772163523215363391094784197813862407788920832021808978329644526664089666789996134473051558272056817878656029829386294494911683664138412872013138428376721
136710886415267266561603062988810286320844845002653696014175835914243307695046656578846633164875973620602887386324225280517042695769624677037752647197469344529688901040734522712143931
1426843545665766686727480960614452366724300825201486069768820520124416231538653265924630773065318470908712562688827268313231367238597740774446063803804928201529141646565425784582792678223072849379749833583241
Technical details
Node id
n-6e3b5b8a1581542e · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d36c4f1a2e9bd6b7

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:      4,       7;
  2:     16,      56,       49;
  3:     64,     336,      588,       343;
  4:    256,    1792,     4704,      5488,      2401;
  5:   1024,    8960,    31360,     54880,     48020,     16807;
  6:   4096,   43008,   188160,    439040,    576240,    403368,    117649;
  7:  16384,  200704,  1053696,   3073280,   5378240,   5647152,   3294172,    823543;
  8:  65536,  917504,  5619712,  19668992,  43025920,  60236288,  52706752,  26353376,   5764801;
  9: 262144, 4128768, 28901376, 118013952, 309786624, 542126592, 632481024, 474360768, 207532836, 40353607;
%F Formula
The row polynomials satisfy P(n,x) = (4+7*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[(4 + 7*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] = (4 + 7*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

Candidate equalities ?

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.