Real-rooted atlas

A109447: Binomial coefficients C(n,k) with n-k odd, read by rows.

Veronese parent of A109447 → Veronese 2 (j=1)

A109447 ↗ nonnegative coefficients

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

Recurrence from A109447

\(P_{n}(x) = \left(2 + 2\,x\right)\,P_{n-2}(x) + \left(-1 + 2\,x - x^{2}\right)\,P_{n-4}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2\)
\(P_{2}(x) = 1 + 3\,x\)
\(P_{3}(x) = 4 + 4\,x\)

Rows ?

nx0x1x2x3Pn(1)
011
122
2134
3448
4110516
5620632
612135764
7856568128

All 15 rows b-files: A000079 (row sums), A034867 (reversed triangle), A109447 ?

Technical details
Node id
n-632227a1ddc9eaa6 · revision 0 · JSON
Definition source
oeis_catalog:A109447
Fingerprint
f6-937db0732045a112

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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

Matches ?

  • A034867 reversed triangle 64 terms, 15 rows · tabf · nonn · catalog b-file?
  • A109447 triangle 64 terms, 15 rows · tabf · nonn · catalog b-file?

Row sums ? A000079, A131577, A220051, A166444, A050732, A118655, A034008, A011782

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  2;
  2:  1,   3;
  3:  4,   4;
  4:  1,  10,   5;
  5:  6,  20,   6;
  6:  1,  21,  35,   7;
  7:  8,  56,  56,   8;
  8:  1,  36, 126,  84,  9;
  9: 10, 120, 252, 120, 10;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-2,x) - (1-2*x+x^2)*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 1 + 3*x, P(3,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 1 + 3*x;
P[3, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] - (1 - 2*x + x^2)*P[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; v[2] = 2; v[3] = 1 + 3*x; v[4] = 4 + 4*x; for(n = 4, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] - (1 - 2*x + x^2)*v[i - 4]); 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

A034867 reversed triangle, offset 0, from P0
%e Example
Triangle begins:
  0:  1;
  1:  2;
  2:  3,   1;
  3:  4,   4;
  4:  5,  10,   1;
  5:  6,  20,   6;
  6:  7,  35,  21,   1;
  7:  8,  56,  56,   8;
  8:  9,  84, 126,  36,  1;
  9: 10, 120, 252, 120, 10;
%F Formula
Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (2+2*x)*P(n-2,x) - (1-2*x+x^2)*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 1 + 3*x, P(3,x) = 4 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2;
P[2, x_] := 1 + 3*x;
P[3, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] - (1 - 2*x + x^2)*P[n - 4, x]];
Table[Reverse[CoefficientList[P[n, x], x]], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 1 + 3*x; v[4] = 4 + 4*x; for(n = 4, N, my(i = n + 1); v[i] = (2 + 2*x)*v[i - 2] - (1 - 2*x + x^2)*v[i - 4]); v};
row(n, p) = {my(c = Vecrev(p)); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A109447 triangle, offset 1, from P0
%e Example
Triangle begins:
   1:  1;
   2:  2;
   3:  1,   3;
   4:  4,   4;
   5:  1,  10,   5;
   6:  6,  20,   6;
   7:  1,  21,  35,   7;
   8:  8,  56,  56,   8;
   9:  1,  36, 126,  84,  9;
  10: 10, 120, 252, 120, 10;
%F Formula
The row polynomials satisfy P(n,x) = (2+2*x)*P(n-2,x) - (1-2*x+x^2)*P(n-4,x) for n >= 5, with P(1,x) = 1, P(2,x) = 2, P(3,x) = 1 + 3*x, P(4,x) = 4 + 4*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2;
P[3, x_] := 1 + 3*x;
P[4, x_] := 4 + 4*x;
P[n_, x_] := P[n, x] = Expand[(2 + 2*x)*P[n - 2, x] - (1 - 2*x + x^2)*P[n - 4, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; v[3] = 1 + 3*x; v[4] = 4 + 4*x; for(n = 5, N, my(i = n); v[i] = (2 + 2*x)*v[i - 2] - (1 - 2*x + x^2)*v[i - 4]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.