A029653: Numbers in (2,1)-Pascal triangle (by row).

Matches ?

• A029653 triangle 70 terms, 11 rows · tabl · nonn · catalog b-file?

Row sums ? A122391, A168680, A167648, A168728, A168776, A168824, A003945, A167881

Submission text ?

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

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) = 2 + x.

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

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) = 2 + x.

P[0, x_] := 1;
P[1, x_] := 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] = 1; v[2] = 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), Iterated real-rootedness preservers, Geometric basis map (elementary), Liu–Wang theorem
• 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, Geometric basis map (elementary), Liu–Wang theorem
• 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, Geometric basis map (elementary)
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
  by Iterated real-rootedness preservers
• Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
  by Geometric basis map (elementary)

Proof chains ? JSON

• Geometric basis map (elementary) polytool ? step ›

  • from Riordan array (1+x, 1) read by rows.?
    • 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
• 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
• Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check

Used in 5

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

Related ?

Same fingerprint ?

• A029635: The (1,2)-Pascal triangle (or Lucas triangle) read by rows.
• A028326 → Pₙ + Pₙ₊₁
• A097805 → basis A013609
• A134239: A127899(unsigned) * A007318.
• A128540 → basis A007318
• A136158 → basis A013611
• A007318 → Pₙ + Pₙ₊₁
• A093560 → basis A013609
• A127952 → basis A013609

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.