{"id":"n-4479b0484a11b442","revision":0,"first_index":1,"fingerprint":"f6-85c07d3149a3188a","name":"Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","1"],["5","6","1"],["14","28","12","1"],["42","120","90","20","1"],["132","495","550","220","30","1"],["429","2002","3003","1820","455","42","1"],["1430","8008","15288","12740","4900","840","56","1"],["4862","31824","74256","79968","42840","11424","1428","72","1"],["16796","125970","348840","465120","325584","122094","23940","2280","90","1"],["58786","497420","1598850","2558160","2238390","1106028","307230","46200","3465","110","1"],["208012","1961256","7191272","13483635","14276790","8883336","3272808","701316","83490","5060","132","1"],["742900","7726160","31870410","68740100","85925125","65122200","30390360","8682960","1480050","143000","7150","156","1"],["2674440","30421755","139582170","341200860","493843350","444459015","253976580","92355120","21081060","2927925","234234","9828","182","1"],["9694845","119759850","605452575","1657028100","2734096365","2864291430","1952925975","873358200","254729475","47549502","5486481","369460","13195","210","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","12","55","273","1428","7752","43263","246675","1430715","8414640","50067108","300830572","1822766520","11124755664"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A120986","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"],["2"]]},"terms":[{"coeff":{"coeffs":[["1","2"],["-11/2","-8"],["15/2","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1/2","-1/2","1"],["1/2","7/2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":170104,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0)","region_exact":null,"edges":["e-ee7174aa33c40ce4"]},{"id":170105,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-ee7174aa33c40ce4"]},{"id":170106,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-ee7174aa33c40ce4"]}],"region_exact":{},"incoming_edges":[{"id":"e-ee7174aa33c40ce4","operation":{"operation":"theorem_check","theorem":"hypergeometric_row"},"operation_detail":"hypergeometric_row","conclusion_id":"n-4479b0484a11b442","premise_count":0,"theorem":"Row-hypergeometric rows: an exact row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, proved for every n by an invariant row-annihilator system of the rule, writes every row as U_0(n) x^v (or its reversal) times the image of (1 + x)^M(n) under a product of nonnegative multiplier sequences: C^k (C > 0); k -> M'!/(M' - k)!, whose exponential generating function is (1 + x)^M' (Polya--Schur, Theorem 7.1 (3) => (1) of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601: a sequence is a multiplier sequence when G(x) = sum_k lambda_k x^k / k! or G(-x) can be written C x^m e^(a x) prod_k (1 + alpha_k x) with a >= 0, alpha_k >= 0); k -> phi(k) for a polynomial phi with only real roots, all <= 0 (Laguerre's theorem, RealRooted isPFMultiplierSequence_eval_of_roots_nonpos); k -> 1/(alpha)_k for alpha > 0 (RealRooted isPFMultiplierSequence_inv_ascPochhammer, via the generalized Laguerre polynomials); products of PF multiplier sequences are PF multiplier sequences (RealRooted IsPFMultiplierSequence.mul) and map real-rooted polynomials with nonnegative coefficients to such polynomials (IsPFMultiplierSequence.diagonalOperator_isPF)","lean_route":null,"lean_declarations":["RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_inv_ascPochhammer","RealRooted.IsPFMultiplierSequence.mul","RealRooted.IsPFMultiplierSequence.diagonalOperator_isPF"],"certificate_hash":"6d61996c0fcb1c75955a24eeed7cdca01eeda5e9e4b8cf2965bd5339a83a72c8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":false,"value":"0"}}}],"premises":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-6f55f8eb0a22acaa","kind":"oeis_match","node_a_id":"n-4479b0484a11b442","node_b_id":null,"fingerprint":null,"oeis_id":"A001764","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001764","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-8e6b096eac5eafb2","kind":"oeis_match","node_a_id":"n-4479b0484a11b442","node_b_id":null,"fingerprint":null,"oeis_id":"A108767","in_catalog":0,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"reversed_triangle","terms_compared":52},"in_catalog":false,"oeis_id":"A108767","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-bedc635986613ce8","kind":"oeis_match","node_a_id":"n-4479b0484a11b442","node_b_id":null,"fingerprint":null,"oeis_id":"A120986","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":9,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":9,"target":"triangle","terms_compared":50},"in_catalog":true,"oeis_id":"A120986","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A120986","relation":"identification"},{"oeis_id":"A108767","relation":"match_reversed_triangle"},{"oeis_id":"A001764","relation":"match_row_sums"},{"oeis_id":"A120986","relation":"match_triangle"}],"bfiles":[{"node_id":"n-4479b0484a11b442","oeis_id":"A108767","target":"reversed_triangle","oeis_offset":1,"first_row":1,"rows_used":44,"terms_matched":52,"oeis_terms_agreeing":52,"oeis_terms_available":52,"term_count":990,"raw_size":21251},{"node_id":"n-4479b0484a11b442","oeis_id":"A120986","target":"triangle","oeis_offset":0,"first_row":1,"rows_used":44,"terms_matched":50,"oeis_terms_agreeing":50,"oeis_terms_available":50,"term_count":990,"raw_size":21249}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:     1;\n   2:     2,      1;\n   3:     5,      6,      1;\n   4:    14,     28,     12,      1;\n   5:    42,    120,     90,     20,      1;\n   6:   132,    495,    550,    220,     30,      1;\n   7:   429,   2002,   3003,   1820,    455,     42,     1;\n   8:  1430,   8008,  15288,  12740,   4900,    840,    56,    1;\n   9:  4862,  31824,  74256,  79968,  42840,  11424,  1428,   72,  1;\n  10: 16796, 125970, 348840, 465120, 325584, 122094, 23940, 2280, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+n)*P(n,x) = ((15/2)*n^2-(11/2)*n+1+(6*n^2-8*n+2)*x)*P(n-1,x) + ((1/2)*n-1/2+((7/2)*n-1/2)*x-(4*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(((15/2)*n^2 - (11/2)*n + 1 + (6*n^2 - 8*n + 2)*x)*P[n - 1, x] + ((1/2)*n - 1/2 + ((7/2)*n - 1/2)*x - (4*n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 + n)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (((15/2)*n^2 - (11/2)*n + 1 + (6*n^2 - 8*n + 2)*x)*v[i - 1] + ((1/2)*n - 1/2 + ((7/2)*n - 1/2)*x - (4*n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 + n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A108767","target":"reversed_triangle","first_row":1,"oeis_offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,  2;\n   3: 1,  6,    5;\n   4: 1, 12,   28,    14;\n   5: 1, 20,   90,   120,     42;\n   6: 1, 30,  220,   550,    495,    132;\n   7: 1, 42,  455,  1820,   3003,   2002,    429;\n   8: 1, 56,  840,  4900,  12740,  15288,   8008,   1430;\n   9: 1, 72, 1428, 11424,  42840,  79968,  74256,  31824,   4862;\n  10: 1, 90, 2280, 23940, 122094, 325584, 465120, 348840, 125970, 16796;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where (2*n^2+n)*P(n,x) = ((15/2)*n^2-(11/2)*n+1+(6*n^2-8*n+2)*x)*P(n-1,x) + ((1/2)*n-1/2+((7/2)*n-1/2)*x-(4*n-1)*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(((15/2)*n^2 - (11/2)*n + 1 + (6*n^2 - 8*n + 2)*x)*P[n - 1, x] + ((1/2)*n - 1/2 + ((7/2)*n - 1/2)*x - (4*n - 1)*x^2)*D[P[n - 1, x], x])/(2*n^2 + n)]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (((15/2)*n^2 - (11/2)*n + 1 + (6*n^2 - 8*n + 2)*x)*v[i - 1] + ((1/2)*n - 1/2 + ((7/2)*n - 1/2)*x - (4*n - 1)*x^2)*deriv(v[i - 1], x))/(2*n^2 + n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c = Vecrev(c); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}},{"oeis_id":"A120986","target":"triangle","first_row":1,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     1;\n  1:     2,      1;\n  2:     5,      6,      1;\n  3:    14,     28,     12,      1;\n  4:    42,    120,     90,     20,      1;\n  5:   132,    495,    550,    220,     30,      1;\n  6:   429,   2002,   3003,   1820,    455,     42,     1;\n  7:  1430,   8008,  15288,  12740,   4900,    840,    56,    1;\n  8:  4862,  31824,  74256,  79968,  42840,  11424,  1428,   72,  1;\n  9: 16796, 125970, 348840, 465120, 325584, 122094, 23940, 2280, 90, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+5*n+3)*P(n,x) = ((15/2)*n^2+(19/2)*n+3+(6*n^2+4*n)*x)*P(n-1,x) + ((1/2)*n+((7/2)*n+3)*x-(4*n+3)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(((15/2)*n^2 + (19/2)*n + 3 + (6*n^2 + 4*n)*x)*P[n - 1, x] + ((1/2)*n + ((7/2)*n + 3)*x - (4*n + 3)*x^2)*D[P[n - 1, x], x])/(2*n^2 + 5*n + 3)]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (((15/2)*n^2 + (19/2)*n + 3 + (6*n^2 + 4*n)*x)*v[i - 1] + ((1/2)*n + ((7/2)*n + 3)*x - (4*n + 3)*x^2)*deriv(v[i - 1], x))/(2*n^2 + 5*n + 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":1,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-4479b0484a11b442","steps":[],"edges":[],"text":"A120986"},"headline":"A120986: Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).","chain":null}