Real-rooted atlas

A038291 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(10\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
240848
360024024864
4800048009606413824
510000080000240003200160207360
6120000012000004800009600096003842985984
71400000016800000840000022400003360002688089641803776
81600000002240000001344000004480000089600001075200716802048573308928
918000000002880000000201600000080640000020160000032256000322560018432046087739670528
10200000000003600000000028800000000134400000004032000000806400000107520000921600046080010240103195607040
1122000000000044000000000039600000000021120000000073920000000177408000002956800000337920000253440001126400225281362182012928
1224000000000005280000000000528000000000031680000000001267200000000354816000000709632000001013760000010137600006758400027033604915217832200896512
132600000000000062400000000000686400000000004576000000000020592000000000658944000000015375360000002635776000003294720000029286400001757184006389760106496231818611654656
142800000000000007280000000000008736000000000006406400000000003203200000000001153152000000003075072000000061501440000009225216000001025024000008200192000447283200149094402293762995809750614016
Technical details
Node id
n-24bd3442a6d1c2f2 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a8e18dd2f0709b33

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:          2;
  2:         40,          8;
  3:        600,        240,         24;
  4:       8000,       4800,        960,        64;
  5:     100000,      80000,      24000,      3200,       160;
  6:    1200000,    1200000,     480000,     96000,      9600,      384;
  7:   14000000,   16800000,    8400000,   2240000,    336000,    26880,     896;
  8:  160000000,  224000000,  134400000,  44800000,   8960000,  1075200,   71680,   2048;
  9: 1800000000, 2880000000, 2016000000, 806400000, 201600000, 32256000, 3225600, 184320, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (10*n+2*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((10*n + 2*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((10*n + 2*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.