A029635: The (1,2)-Pascal triangle (or Lucas triangle) read by rows.

Matches ?

• A029635 triangle 76 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A168680, A167648, A042950, A007283, A003945, A122391, A098011, A167881

Submission text ?

Triangle begins:
  0: 2;
  1: 1,  2;
  2: 1,  3,  2;
  3: 1,  4,  5,   2;
  4: 1,  5,  9,   7,   2;
  5: 1,  6, 14,  16,   9,   2;
  6: 1,  7, 20,  30,  25,  11,   2;
  7: 1,  8, 27,  50,  55,  36,  13,  2;
  8: 1,  9, 35,  77, 105,  91,  49, 15,  2;
  9: 1, 10, 44, 112, 182, 196, 140, 64, 17, 2;

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) = 2, P(1,x) = 1 + 2*x.

P[0, x_] := 2;
P[1, x_] := 1 + 2*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

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 1 + 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)

Triangle begins:
  0: 2;
  1: 1,  2;
  2: 1,  3,  2;
  3: 1,  4,  5,   2;
  4: 1,  5,  9,   7,   2;
  5: 1,  6, 14,  16,   9,   2;
  6: 1,  7, 20,  30,  25,  11,   2;
  7: 1,  8, 27,  50,  55,  36,  13,  2;
  8: 1,  9, 35,  77, 105,  91,  49, 15,  2;
  9: 1, 10, 44, 112, 182, 196, 140, 64, 17, 2;

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) = 2, P(1,x) = 1 + 2*x.

P[0, x_] := 2;
P[1, x_] := 1 + 2*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

P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 1 + 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)

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), Liu–Wang theorem
• Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
  derived from Row-hypergeometric rows (multiplier sequences), Liu–Wang theorem, 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), Liu–Wang theorem, Iterated real-rootedness preservers
• 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

• 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
• Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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

• basis_map_A013609 of The (1,2)-Pascal triangle (or Lucas triangle) read by rows. Image under a basis map u^k -> P_k (A013609)
• basis_map_A008277 of The (1,2)-Pascal triangle (or Lucas triangle) read by rows. Image under a basis map u^k -> P_k (A008277)
• basis_map_A013611 of The (1,2)-Pascal triangle (or Lucas triangle) read by rows. Image under a basis map u^k -> P_k (A013611)
• basis_map_A013610 of The (1,2)-Pascal triangle (or Lucas triangle) read by rows. Image under a basis map u^k -> P_k (A013610)
• basis_map_A007318 of The (1,2)-Pascal triangle (or Lucas triangle) read by rows. Image under a basis map u^k -> P_k (A007318)

Related ?

Same fingerprint ?

• A029653: Numbers in (2,1)-Pascal triangle (by row).
• A007318 → Pₙ + Pₙ₊₁
• A097805 → basis A013609
• A093560 → basis A013609
• A128540 → basis A007318
• A127952 → basis A013609
• A028326 → Pₙ + Pₙ₊₁
• A134239: A127899(unsigned) * A007318.
• A136158 → basis A013611

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