Real-rooted atlas

A038303 → basis A007318

nonnegative coefficients only rational roots

Proved for all n: all roots real, in (−∞, 0] · coefficients of one sign · Hadamard kernel what this means

Recurrence
\(P_{n}(x) = \left(11 + x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111112
2121221144
313313633311728
414641532472644120736
516105173205133101210551248832
617715619663062196152662018156612985984
71948717112400927338207151243546585254177135831808
82143588811558973684960370890188561024870745363388881429981696
92357947691192922992970153815614881112420292426184476611180443569915159780352
102593742460123579476910964614964523384605203720278104058485230746101597205445110161917364224
1128531167061128531167061112968712300535369215365643076643081846118274405562483153021961566551211743008370688
123138428376721342374004733217118700236665187484920201061076460951543383943216369223641275523927247295292820798613218916100448256
13345227121439314079956889737322254310307658741810343588616859325990652758798798473343998543630399986762072726371046831538066694381431106993205379072
14379749833583241483317970015034285596982281611103853448102404259633620256014720611277382643719719643668799708725319997683322424102146556414844841101115411283918464548864

Generated b-files: A001021 (row sums), A013618 (reversed triangle), A038315 ?

Computed in your browser ?

Technical details
Node id
n-31f0871ac8eccdd6 · revision 1 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-b6561930bb158c66

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

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:         11,          1;
  2:        121,         22,         1;
  3:       1331,        363,        33,         1;
  4:      14641,       5324,       726,        44,        1;
  5:     161051,      73205,     13310,      1210,       55,       1;
  6:    1771561,     966306,    219615,     26620,     1815,      66,      1;
  7:   19487171,   12400927,   3382071,    512435,    46585,    2541,     77,    1;
  8:  214358881,  155897368,  49603708,   9018856,  1024870,   74536,   3388,   88,  1;
  9: 2357947691, 1929229929, 701538156, 148811124, 20292426, 1844766, 111804, 4356, 99, 1;
%F Formula
The row polynomials satisfy P(n,x) = (11+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[(11 + 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] = (11 + 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)

Generated with polytool c48629374597.

A013618: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0: 1;
  1: 1, 11;
  2: 1, 22,  121;
  3: 1, 33,  363,   1331;
  4: 1, 44,  726,   5324,   14641;
  5: 1, 55, 1210,  13310,   73205,   161051;
  6: 1, 66, 1815,  26620,  219615,   966306,   1771561;
  7: 1, 77, 2541,  46585,  512435,  3382071,  12400927,  19487171;
  8: 1, 88, 3388,  74536, 1024870,  9018856,  49603708, 155897368,  214358881;
  9: 1, 99, 4356, 111804, 1844766, 20292426, 148811124, 701538156, 1929229929, 2357947691;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (11+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[(11 + x)*P[n - 1, x]];
Table[Reverse[PadRight[CoefficientList[P[n, x], x], n + 1]], {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] = (11 + x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

A038315: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         11,          1;
  2:        121,         22,         1;
  3:       1331,        363,        33,         1;
  4:      14641,       5324,       726,        44,        1;
  5:     161051,      73205,     13310,      1210,       55,       1;
  6:    1771561,     966306,    219615,     26620,     1815,      66,      1;
  7:   19487171,   12400927,   3382071,    512435,    46585,    2541,     77,    1;
  8:  214358881,  155897368,  49603708,   9018856,  1024870,   74536,   3388,   88,  1;
  9: 2357947691, 1929229929, 701538156, 148811124, 20292426, 1844766, 111804, 4356, 99, 1;
%F Formula
The row polynomials satisfy P(n,x) = (11+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[(11 + x)*P[n - 1, x]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {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] = (11 + x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

  • 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-34d290d6207a0155
  • 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-34d290d6207a0155
  • 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-34d290d6207a0155

Proof chains

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

Same fingerprint (1)

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