A380860: Triangle read by rows: T(n,m) (0<=m<=n) = number of positive n-digit numbers that have exactly m copies of a specific, previously selected positive base-10 digit among its digits.

A380860 ↗ nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence from A380860

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9 - x + 9\,n + n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 8 + x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	8	1							9
2	72	17	1						90
3	648	225	26	1					900
4	5832	2673	459	35	1				9000
5	52488	29889	6804	774	44	1			90000
6	472392	321489	91125	13770	1170	53	1		900000
7	4251528	3365793	1141614	215055	24300	1647	62	1	9000000

All 15 rows b-files: A052268 (row sums), A380860 ?

Technical details

Node id: n-3c85c2a2c3ff061d · revision 1 · JSON
Definition source: oeis_catalog:A380860
Fingerprint: f6-beba9ed1d4d6e664

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 ?

A380860 triangle 55 terms, 10 rows · tabl · nonn · catalog b-file?

Row sums ? A052268, A063945

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         1;
  1:         8,         1;
  2:        72,        17,         1;
  3:       648,       225,        26,        1;
  4:      5832,      2673,       459,       35,       1;
  5:     52488,     29889,      6804,      774,      44,      1;
  6:    472392,    321489,     91125,    13770,    1170,     53,     1;
  7:   4251528,   3365793,   1141614,   215055,   24300,   1647,    62,    1;
  8:  38263752,  34543665,  13640319,  3077109,  433755,  39123,  2205,   71,  1;
  9: 344373768, 349156737, 157306536, 41334300, 6980904, 785862, 58968, 2844, 80, 1;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (9*n-9+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 8 + x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n - 9 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];
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] = 8 + x; for(n = 2, N, my(i = n + 1); v[i] = ((9*n - 9 + (n - 1)*x)*v[i - 1])/(n - 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)

polytool c48629374597

A380860 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0:         1;
  1:         8,         1;
  2:        72,        17,         1;
  3:       648,       225,        26,        1;
  4:      5832,      2673,       459,       35,       1;
  5:     52488,     29889,      6804,      774,      44,      1;
  6:    472392,    321489,     91125,    13770,    1170,     53,     1;
  7:   4251528,   3365793,   1141614,   215055,   24300,   1647,    62,    1;
  8:  38263752,  34543665,  13640319,  3077109,  433755,  39123,  2205,   71,  1;
  9: 344373768, 349156737, 157306536, 41334300, 6980904, 785862, 58968, 2844, 80, 1;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (9*n-9+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 8 + x;
P[n_, x_] := P[n, x] = Expand[Together[((9*n - 9 + (n - 1)*x)*P[n - 1, x])/(n - 1)]];
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; v[2] = 8 + x; for(n = 2, N, my(i = n + 1); v[i] = ((9*n - 9 + (n - 1)*x)*v[i - 1])/(n - 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)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Liu–Wang theorem, Row-hypergeometric rows (multiplier sequences)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Iterated real-rootedness preservers, Liu–Wang theorem, Row-hypergeometric rows (multiplier sequences)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Iterated real-rootedness preservers, Liu–Wang theorem, Row-hypergeometric rows (multiplier sequences)
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 2
by Liu–Wang theorem
Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Iterated real-rootedness preservers
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Iterated real-rootedness preservers

Proof chains ? JSON

Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

References

[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
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.