Real-rooted atlas

A038291 → basis A013611 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(5 + 2\,n x\right)\,P_{n}(x) = \left(50 + 40\,x + 8\,x^{2} + 20\,n x + 8\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110818
210016048308
31000240014402565096
410000320002880010240128082320
51000004000004800002560006400061441306144
6100000048000007200000512000019200003686402867220437312
710000000560000001008000008960000044800000129024002007040131072316240512
81000000006400000001344000000143360000089600000034406400080281600104857605898244849021184
9100000000072000000001728000000021504000000161280000007741440000240844800047185920053084160262144073789452800
10100000000008000000000021600000000030720000000026880000000015482880000060211200000157286400002654208000262144000115343361115696526336
1110000000000088000000000026400000000004224000000000422400000000028385280000001324646400000432537600000973209600001441792000012687769605033164816776769988608
12100000000000096000000000003168000000000056320000000000633600000000004866048000000026492928000000103809024000002919628800000576716800000761266176006039797760218103808251073040519168
131000000000000010400000000000037440000000000073216000000000091520000000000079073280000000049201152000000022491955200000075910348800000187432960000003298820096000392586854400283534950409395240963741798216769536
141000000000000001120000000000000436800000000000093184000000000001281280000000000012300288000000000861020160000000044983910400000001771241472000000524812288000000115458703360000183207198720001984744652800131533373440402653184055560034127790080
Technical details
Node id
n-b82a18a2f7e9534b · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-65a6331fbe5d08da

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:         10,          8;
  2:        100,        160,          48;
  3:       1000,       2400,        1440,         256;
  4:      10000,      32000,       28800,       10240,        1280;
  5:     100000,     400000,      480000,      256000,       64000,       6144;
  6:    1000000,    4800000,     7200000,     5120000,     1920000,     368640,      28672;
  7:   10000000,   56000000,   100800000,    89600000,    44800000,   12902400,    2007040,    131072;
  8:  100000000,  640000000,  1344000000,  1433600000,   896000000,  344064000,   80281600,  10485760,   589824;
  9: 1000000000, 7200000000, 17280000000, 21504000000, 16128000000, 7741440000, 2408448000, 471859200, 53084160, 2621440;
%F Formula
The row polynomials satisfy (5+2*n*x)*P(n,x) = (50+(20*n+40)*x+(8*n+8)*x^2)*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[Together[((50 + (20*n + 40)*x + (8*n + 8)*x^2)*P[n - 1, x])/(5 + 2*n*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] = ((50 + (20*n + 40)*x + (8*n + 8)*x^2)*v[i - 1])/(5 + 2*n*x)); 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.