Real-rooted atlas

A137597: Triangle read by rows: A008277 * A007318.

A007318 → basis A049020

A137597 nonnegative coefficients

Proved for all n: all roots real, in (−∞, −1] · coefficients of one sign · interlacing (weak) · Hadamard kernel what this means

Recurrence from OEIS A137597
\(P_{n}(x) = \left(2 + x\right)\,P_{n-1}(x) + \left(1 + x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
255111
315229147
4529961141227
52034713851352011215
68772386241611402602717107
7414012867154709156283545535144959
8211477368110221572590284416230742441305091
9115975446620700311578929274386765031247411465412206399
10678570285645749846514692500260776088475118431723220169565116913987
1142135971921724336872561388781912476281599069192521827406857407552420771136823263
12276444371356104482833420603303147762369896761092422103292953465095808361876814533559011163490499
1319089932210011599012259923016288287464422984638821199262350418245542982052501548475516193321093954537104110366252031
141382958545771422505718689475897258681517642266996968613202773584523266343614259708322676267883439135729819791696246006119196491364675

Generated b-files: A126350 (reversed triangle), A137597 ?

Computed in your browser ?

Technical details
Node id
n-bdae5aa3d1cac0ec · revision 2 · JSON
Definition source
oeis_catalog:A137597
Fingerprint
f6-1090d35600282d8c

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): no

OEIS matches ?

A generated b-file is computed from the node's definition along the row alignment of its OEIS link and starts at the entry's offset. It is an identification on a finite prefix, not a proof that the node and the OEIS entry agree for every n.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      2,      1;
  2:      5,      5,      1;
  3:     15,     22,      9,      1;
  4:     52,     99,     61,     14,      1;
  5:    203,    471,    385,    135,     20,     1;
  6:    877,   2386,   2416,   1140,    260,    27,     1;
  7:   4140,  12867,  15470,   9156,   2835,   455,    35,    1;
  8:  21147,  73681, 102215,  72590,  28441,  6230,   742,   44,  1;
  9: 115975, 446620, 700311, 578929, 274386, 76503, 12474, 1146, 54, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (1+x)*d/dx 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[(2 + x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], 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] = (2 + x)*v[i - 1] + (1 + x)*deriv(v[i - 1], 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)

Generated with polytool c48629374597.

A126350: reversed triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1: 1;
   2: 1,  2;
   3: 1,  5,    5;
   4: 1,  9,   22,    15;
   5: 1, 14,   61,    99,    52;
   6: 1, 20,  135,   385,   471,    203;
   7: 1, 27,  260,  1140,  2416,   2386,    877;
   8: 1, 35,  455,  2835,  9156,  15470,  12867,   4140;
   9: 1, 44,  742,  6230, 28441,  72590, 102215,  73681,  21147;
  10: 1, 54, 1146, 12474, 76503, 274386, 578929, 700311, 446620, 115975;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2 + x)*v[i - 1] + (1 + x)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

A137597: triangle, offset 1, from the row P0
%e Example
Triangle begins:
   1:      1;
   2:      2,      1;
   3:      5,      5,      1;
   4:     15,     22,      9,      1;
   5:     52,     99,     61,     14,      1;
   6:    203,    471,    385,    135,     20,     1;
   7:    877,   2386,   2416,   1140,    260,    27,     1;
   8:   4140,  12867,  15470,   9156,   2835,   455,    35,    1;
   9:  21147,  73681, 102215,  72590,  28441,  6230,   742,   44,  1;
  10: 115975, 446620, 700311, 578929, 274386, 76503, 12474, 1146, 54, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2+x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2 + x)*v[i - 1] + (1 + x)*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Generated with polytool c48629374597.

Proved properties

  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-3c112fda27f20cb0, e-62a7be97cd0c2a0d
  • Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n have one sign
    All coefficients of each polynomial have the same sign (after x -> -x when alternating).
    derived from the cited steps by a stated theorem: e-3c112fda27f20cb0, e-62a7be97cd0c2a0d, e-9acfc4c7c0e80a75
  • Hadamard kernel (Malo). Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)
    Multiplying the coefficients of any real-rooted polynomial termwise by those of a row (Hadamard product) gives a real-rooted polynomial again.
    derived from the cited steps by a stated theorem: e-3c112fda27f20cb0, e-62a7be97cd0c2a0d, e-9acfc4c7c0e80a75
  • Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    The roots of consecutive polynomials alternate on the real line.
    by the cited steps: e-62a7be97cd0c2a0d
  • Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
    Every polynomial of the sequence has only real roots, inside the stated interval.
    by the cited steps: e-9acfc4c7c0e80a75

Proof chains

3 incoming proof steps. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

Used in 7 proof steps

Same fingerprint (3)

Equal up to content, sign, reversal and a small index shift.

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. [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
  4. [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
  5. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  6. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.