A097805: Number of compositions of n with k parts, T(n, k) = binomial(n-1, k-1) for n, k >= 1 and T(n, 0) = 0^n, triangle read by rows for n >= 0 and 0 <= k <= n.

A097805 ↗ palindromicγ-positivenonnegative coefficientsonly rational roots

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

Recurrence from A097805

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

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	1														1
2	0	1	1													2
3	0	1	2	1												4
4	0	1	3	3	1											8
5	0	1	4	6	4	1										16
6	0	1	5	10	10	5	1									32
7	0	1	6	15	20	15	6	1								64
8	0	1	7	21	35	35	21	7	1							128
9	0	1	8	28	56	70	56	28	8	1						256
10	0	1	9	36	84	126	126	84	36	9	1					512
11	0	1	10	45	120	210	252	210	120	45	10	1				1024
12	0	1	11	55	165	330	462	462	330	165	55	11	1			2048
13	0	1	12	66	220	495	792	924	792	495	220	66	12	1		4096
14	0	1	13	78	286	715	1287	1716	1716	1287	715	286	78	13	1	8192

b-files: A007318, A007318 (reversed triangle), A011782 (row sums), A071919 (reversed triangle), A097805 ?

Technical details

Node id: n-3783f6df357ea675 · revision 1 · JSON
Definition source: oeis_catalog:A097805
Fingerprint: f6-77a7ed3b1d5f257e

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: yes
γ-positive (gamma_positive): yes
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

A071919 reversed triangle 91 terms, 13 rows · tabl · nonn · catalog b-file?
A007318 reversed triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?
A007318 triangle 78 terms, 12 rows · tabl · nonn · catalog b-file?
A097805 triangle 91 terms, 13 rows · tabl · nonn · catalog b-file?

Row sums ? A166444, A220469, A274866, A249169, A274864, A011782, A220493, A274865

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1, 1;
  3: 0, 1, 2,  1;
  4: 0, 1, 3,  3,  1;
  5: 0, 1, 4,  6,  4,  1;
  6: 0, 1, 5, 10, 10,  5,  1;
  7: 0, 1, 6, 15, 20, 15,  6,  1;
  8: 0, 1, 7, 21, 35, 35, 21,  7, 1;
  9: 0, 1, 8, 28, 56, 70, 56, 28, 8, 1;

%F Formula

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

%t Mathematica

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

A007318 triangle, offset 0, from P₁, low zeros dropped

%e Example

Triangle begins:
  0: 1;
  1: 1, 1;
  2: 1, 2,  1;
  3: 1, 3,  3,  1;
  4: 1, 4,  6,  4,   1;
  5: 1, 5, 10, 10,   5,   1;
  6: 1, 6, 15, 20,  15,   6,  1;
  7: 1, 7, 21, 35,  35,  21,  7,  1;
  8: 1, 8, 28, 56,  70,  56, 28,  8, 1;
  9: 1, 9, 36, 84, 126, 126, 84, 36, 9, 1;

%F Formula

Row n lists the coefficients of P(n,x) in ascending powers of x, starting at the lowest nonzero coefficient, where n*P(n,x) = (n+n*x)*P(n-1,x) for n >= 1, with P(0,x) = x.

%t Mathematica

P[0, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[((n + n*x)*P[n - 1, x])/(n)]];
Table[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = x; for(n = 1, N, my(i = n + 1); v[i] = ((n + n*x)*v[i - 1])/(n)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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

A007318 reversed triangle, offset 0, from P₁, low zeros dropped

%e Example

Triangle begins:
  0: 1;
  1: 1, 1;
  2: 1, 2,  1;
  3: 1, 3,  3,  1;
  4: 1, 4,  6,  4,   1;
  5: 1, 5, 10, 10,   5,   1;
  6: 1, 6, 15, 20,  15,   6,  1;
  7: 1, 7, 21, 35,  35,  21,  7,  1;
  8: 1, 8, 28, 56,  70,  56, 28,  8, 1;
  9: 1, 9, 36, 84, 126, 126, 84, 36, 9, 1;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where n*P(n,x) = (n+n*x)*P(n-1,x) for n >= 1, with P(0,x) = x.

%t Mathematica

P[0, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[((n + n*x)*P[n - 1, x])/(n)]];
Table[Reverse[PadRight[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x], n + 1]], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = x; for(n = 1, N, my(i = n + 1); v[i] = ((n + n*x)*v[i - 1])/(n)); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); 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)

polytool c48629374597

A071919 reversed triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 1, 0;
  2: 1, 1,  0;
  3: 1, 2,  1,  0;
  4: 1, 3,  3,  1,  0;
  5: 1, 4,  6,  4,  1,  0;
  6: 1, 5, 10, 10,  5,  1,  0;
  7: 1, 6, 15, 20, 15,  6,  1, 0;
  8: 1, 7, 21, 35, 35, 21,  7, 1, 0;
  9: 1, 8, 28, 56, 70, 56, 28, 8, 1, 0;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, where (n-1)*P(n,x) = (n-1+(n-1)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = x.

%t Mathematica

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

polytool c48629374597

A097805 triangle, offset 0, from P₀

%e Example

Triangle begins:
  0: 1;
  1: 0, 1;
  2: 0, 1, 1;
  3: 0, 1, 2,  1;
  4: 0, 1, 3,  3,  1;
  5: 0, 1, 4,  6,  4,  1;
  6: 0, 1, 5, 10, 10,  5,  1;
  7: 0, 1, 6, 15, 20, 15,  6,  1;
  8: 0, 1, 7, 21, 35, 35, 21,  7, 1;
  9: 0, 1, 8, 28, 56, 70, 56, 28, 8, 1;

%F Formula

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

%t Mathematica

P[0, x_] := 1;
P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1 + (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] = x; for(n = 2, N, my(i = n + 1); v[i] = ((n - 1 + (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 Row-hypergeometric rows (multiplier sequences), Iterated real-rootedness preservers
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Row-hypergeometric rows (multiplier sequences), Iterated real-rootedness preservers
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Row-hypergeometric rows (multiplier sequences), 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

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
Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check

Used in 5

Related ?

Same fingerprint ?

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