A136521: Triangle read by rows: (1, 2, 2, 2, ...) on the main diagonal and the rest zeros.

Matches ?

• A136521 triangle 91 terms, 13 rows · tabl · nonn · catalog b-file?

Submission text ?

Triangle begins:
  0: 1;
  1: 0, 2;
  2: 0, 0, 2;
  3: 0, 0, 0, 2;
  4: 0, 0, 0, 0, 2;
  5: 0, 0, 0, 0, 0, 2;
  6: 0, 0, 0, 0, 0, 0, 2;
  7: 0, 0, 0, 0, 0, 0, 0, 2;
  8: 0, 0, 0, 0, 0, 0, 0, 0, 2;
  9: 0, 0, 0, 0, 0, 0, 0, 0, 0, 2;

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

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1)*x*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((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)

Triangle begins:
  0: 1;
  1: 0, 2;
  2: 0, 0, 2;
  3: 0, 0, 0, 2;
  4: 0, 0, 0, 0, 2;
  5: 0, 0, 0, 0, 0, 2;
  6: 0, 0, 0, 0, 0, 0, 2;
  7: 0, 0, 0, 0, 0, 0, 0, 2;
  8: 0, 0, 0, 0, 0, 0, 0, 0, 2;
  9: 0, 0, 0, 0, 0, 0, 0, 0, 0, 2;

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

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n - 1)*x*P[n - 1, x])/(n - 1)]];
Table[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((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)

Properties

• Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
  by Iterated real-rootedness preservers
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Iterated real-rootedness preservers
• Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
  derived from 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

Related ?

Same fingerprint ?

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• A198954: Expansion of the rotational partition function for a heteronuclear diatomic molecule.
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• A038325 → γ
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• A127773 → (θ+1)²
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• hadamard_factor parent of Triangle read by rows: row n consists of n-1 zeros followed by n(n+1)/2.
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References

1. [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
2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.