Real-rooted atlas

A038291 → basis A013609

nonnegative coefficients only rational roots

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

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
110212
2100404144
3100060012081728
410000800024003201620736
510000010000040000800080032248832
610000001200000600000160000240001920642985984
710000000140000008400000280000056000067200448012835831808
81000000001600000001120000004480000011200000179200017920010240256429981696
9100000000018000000001440000000672000000201600000403200005376000460800230405125159780352
101000000000020000000000180000000009600000000336000000080640000013440000015360000115200051200102461917364224
11100000000000220000000000220000000000132000000000528000000001478400000029568000004224000004224000028160001126402048743008370688
12100000000000024000000000002640000000000176000000000079200000000025344000000059136000000101376000001267200000112640000675840024576040968916100448256
13100000000000002600000000000031200000000000228800000000001144000000000041184000000001098240000000219648000000329472000003660800000292864000159744005324808192106993205379072
141000000000000002800000000000003640000000000002912000000000001601600000000006406400000000019219200000000439296000000076876800000010250240000010250240000745472000372736001146880163841283918464548864

Generated b-files: A001021 (row sums), A038216 (reversed triangle), A038304 ?

Computed in your browser ?

Technical details
Node id
n-6323575beb222864 · revision 1 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-a8e18dd2f0709b33

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:         10,          2;
  2:        100,         40,          4;
  3:       1000,        600,        120,         8;
  4:      10000,       8000,       2400,       320,        16;
  5:     100000,     100000,      40000,      8000,       800,       32;
  6:    1000000,    1200000,     600000,    160000,     24000,     1920,      64;
  7:   10000000,   14000000,    8400000,   2800000,    560000,    67200,    4480,    128;
  8:  100000000,  160000000,  112000000,  44800000,  11200000,  1792000,  179200,  10240,   256;
  9: 1000000000, 1800000000, 1440000000, 672000000, 201600000, 40320000, 5376000, 460800, 23040, 512;
%F Formula
The row polynomials satisfy P(n,x) = (10+2*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[(10 + 2*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] = (10 + 2*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.

A038216: reversed triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:   1;
  1:   2,    10;
  2:   4,    40,    100;
  3:   8,   120,    600,    1000;
  4:  16,   320,   2400,    8000,    10000;
  5:  32,   800,   8000,   40000,   100000,    100000;
  6:  64,  1920,  24000,  160000,   600000,   1200000,   1000000;
  7: 128,  4480,  67200,  560000,  2800000,   8400000,  14000000,   10000000;
  8: 256, 10240, 179200, 1792000, 11200000,  44800000, 112000000,  160000000,  100000000;
  9: 512, 23040, 460800, 5376000, 40320000, 201600000, 672000000, 1440000000, 1800000000, 1000000000;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (10+2*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[(10 + 2*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] = (10 + 2*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.

A038304: triangle, offset 0, from the row P0
%e Example
Triangle begins:
  0:          1;
  1:         10,          2;
  2:        100,         40,          4;
  3:       1000,        600,        120,         8;
  4:      10000,       8000,       2400,       320,        16;
  5:     100000,     100000,      40000,      8000,       800,       32;
  6:    1000000,    1200000,     600000,    160000,     24000,     1920,      64;
  7:   10000000,   14000000,    8400000,   2800000,    560000,    67200,    4480,    128;
  8:  100000000,  160000000,  112000000,  44800000,  11200000,  1792000,  179200,  10240,   256;
  9: 1000000000, 1800000000, 1440000000, 672000000, 201600000, 40320000, 5376000, 460800, 23040, 512;
%F Formula
The row polynomials satisfy P(n,x) = (10+2*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[(10 + 2*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] = (10 + 2*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-88ca34e9877f4f47
  • 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-88ca34e9877f4f47
  • 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-88ca34e9877f4f47

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 (5)

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