Real-rooted atlas

A038291 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4 + x + n\right)\,P_{n}(x) = \left(50 + 20\,x + 2\,x^{2} + 10\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
112214
2140484192
3160084014482592
4180001280033603841634560
5200000180000640001120096032456192
6220000024000001080000256000336002304645971968
7240000003080000016800000504000089600094080537612877635584
826000000038400000024640000089600000201600002867200250880122882561003290624
928000000004680000000345600000014784000004032000007257600086016006451202764851212899450880
103000000000056000000000468000000002304000000073920000001612800000241920000245760001612800614401024165112971264
1132000000000066000000000061600000000034320000000012672000000032524800000591360000076032000067584000394240013516820482105190383616
12340000000000076800000000007920000000000492800000000020592000000006082560000001300992000002027520000022809600001802240009461760294912409626748301344768
133600000000000088400000000000998400000000006864000000000032032000000000107078400000002635776000000483225600000658944000006589440000468582400223641606389768192338811817033728
143800000000000001008000000000000123760000000000093184000000000048048000000000017937920000000049969920000000105431040000001691289600000205004800000184504320001192755200521830401376256163844279728215162880
Technical details
Node id
n-fc60b80193fbbe49 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d5fd215a74ece842

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:         12,          2;
  2:        140,         48,          4;
  3:       1600,        840,        144,          8;
  4:      18000,      12800,       3360,        384,        16;
  5:     200000,     180000,      64000,      11200,       960,       32;
  6:    2200000,    2400000,    1080000,     256000,     33600,     2304,      64;
  7:   24000000,   30800000,   16800000,    5040000,    896000,    94080,    5376,    128;
  8:  260000000,  384000000,  246400000,   89600000,  20160000,  2867200,  250880,  12288,   256;
  9: 2800000000, 4680000000, 3456000000, 1478400000, 403200000, 72576000, 8601600, 645120, 27648, 512;
%F Formula
The row polynomials satisfy (n+4+x)*P(n,x) = (10*n+50+(2*n+20)*x+2*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[((10*n + 50 + (2*n + 20)*x + 2*x^2)*P[n - 1, x])/(n + 4 + 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] = ((10*n + 50 + (2*n + 20)*x + 2*x^2)*v[i - 1])/(n + 4 + 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.