{"id":"n-5939aa09e8ea2fc4","revision":1,"first_index":0,"fingerprint":"f6-09fed58a54c64e77","name":"Coefficient triangle of certain polynomials N(5; m,x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1","6"],["1","14","21"],["1","24","84","56"],["1","36","216","336","126"],["1","50","450","1200","1050","252"],["1","66","825","3300","4950","2772","462"],["1","84","1386","7700","17325","16632","6468","792"],["1","104","2184","16016","50050","72072","48048","13728","1287"],["1","126","3276","30576","126126","252252","252252","123552","27027","2002"],["1","150","4725","54600","286650","756756","1051050","772200","289575","50050","3003"],["1","176","6600","92400","600600","2018016","3699696","3775200","2123550","629200","88088","4368"],["1","204","8976","149600","1178100","4900896","11435424","15402816","12033450","5348200","1283568","148512","6188"],["1","234","11934","233376","2187900","11027016","31855824","54609984","56316546","34763300","12514788","2482272","241332","8568"],["1","266","15561","352716","3879876","23279256","81477396","172931616","226972746","184940756","92470378","27511848","4585308","379848","11628"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","36","165","715","3003","12376","50388","203490","817190","3268760","13037895","51895935","206253075","818809200"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A062190","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["5"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","20"],["5","13"],["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","5","-5"],["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":158790,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0)","region_exact":null,"edges":["e-e968c429d4b85e0b"]},{"id":158791,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-e968c429d4b85e0b"]},{"id":158792,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-e968c429d4b85e0b"]},{"id":158793,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 1","region_exact":null,"edges":["e-e968c429d4b85e0b"]}],"region_exact":{},"incoming_edges":[{"id":"e-e968c429d4b85e0b","operation":{"operation":"theorem_check","theorem":"liu_wang"},"operation_detail":"liu_wang","conclusion_id":"n-5939aa09e8ea2fc4","premise_count":0,"theorem":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","lean_route":null,"lean_declarations":[],"certificate_hash":"c066ec3c296acba272cd9f5bbfb798172bf416303e9dd809faf9b6f915c0c862","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":false,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[]}],"used_by":[{"id":"e-581332e7358bcae7","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-dae9c9c9d6ccc7dc","conclusion_name":"derivative of Coefficient triangle of certain polynomials N(5; m,x)."},{"id":"e-595b0519ab8ba235","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-27ff36c5792a7014","conclusion_name":"theta_plus_one of Coefficient triangle of certain polynomials N(5; m,x)."},{"id":"e-725543b0eb30a334","operation":"preserver_image","operation_detail":"mobius, shift 0","conclusion_id":"n-249188ec3e0ce4f3","conclusion_name":"mobius_x_over_x_plus_one of Coefficient triangle of certain polynomials N(5; m,x)."},{"id":"e-888afb9e2aff2a08","operation":"preserver_image","operation_detail":"reversal, shift 0","conclusion_id":"n-817c504263a54deb","conclusion_name":"reversal of Coefficient triangle of certain polynomials N(5; m,x)."},{"id":"e-e91167a9e27d414d","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-7b2cc3bab513d84f","conclusion_name":"shift_x_plus_one of Coefficient triangle of certain polynomials N(5; m,x)."},{"id":"e-f9926095268ea913","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-c864ab35c5b8f362","conclusion_name":"one_plus_derivative of Coefficient triangle of certain polynomials N(5; m,x)."}],"used_by_total":6,"links":[{"id":"l-baf89cf853bc1e18","kind":"fingerprint_related","node_a_id":"n-5939aa09e8ea2fc4","node_b_id":"n-817c504263a54deb","fingerprint":"f6-09fed58a54c64e77","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-09fed58a54c64e77"},"other_id":"n-817c504263a54deb","other":{"id":"n-817c504263a54deb","name":"reversal of Coefficient triangle of certain polynomials N(5; m,x).","first_index":0,"fingerprint":"f6-09fed58a54c64e77","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","1"],["21","14","1"]],"card_rows":[["1"],["6","1"],["21","14","1"],["56","84","24","1"]],"row_sums_preview":["1","7","36","165","715","3003","12376","50388"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5939aa09e8ea2fc4","steps":["reverse"],"edges":["e-888afb9e2aff2a08"],"text":"A062190 \u2192 reverse"},"headline":"A062190 \u2192 reverse","chain":null}},{"id":"l-71392d186d5df0fd","kind":"oeis_match","node_a_id":"n-5939aa09e8ea2fc4","node_b_id":null,"fingerprint":null,"oeis_id":"A003516","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":2,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A003516","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-bf1f574931226c9e","kind":"oeis_match","node_a_id":"n-5939aa09e8ea2fc4","node_b_id":null,"fingerprint":null,"oeis_id":"A062190","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A062190","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A062190","relation":"identification"},{"oeis_id":"A003516","relation":"match_row_sums"},{"oeis_id":"A062190","relation":"match_triangle"}],"bfiles":[{"node_id":"n-5939aa09e8ea2fc4","oeis_id":"A003516","target":"row_sums","oeis_offset":2,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":22,"oeis_terms_available":22,"term_count":200,"raw_size":13033},{"node_id":"n-5939aa09e8ea2fc4","oeis_id":"A062190","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":17564}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1,   6;\n  2: 1,  14,   21;\n  3: 1,  24,   84,    56;\n  4: 1,  36,  216,   336,    126;\n  5: 1,  50,  450,  1200,   1050,    252;\n  6: 1,  66,  825,  3300,   4950,   2772,    462;\n  7: 1,  84, 1386,  7700,  17325,  16632,   6468,    792;\n  8: 1, 104, 2184, 16016,  50050,  72072,  48048,  13728,  1287;\n  9: 1, 126, 3276, 30576, 126126, 252252, 252252, 123552, 27027, 2002;\n","recurrence_sentence":"The row polynomials satisfy (n^2+5*n)*P(n,x) = (n^2+5*n+(3*n^2+13*n+20)*x)*P(n-1,x) + ((2*n+5)*x-(2*n+5)*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[((n^2 + 5*n + (3*n^2 + 13*n + 20)*x)*P[n - 1, x] + ((2*n + 5)*x - (2*n + 5)*x^2)*D[P[n - 1, x], x])/(n^2 + 5*n)]];\nTable[CoefficientList[P[n, x], x], {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] = ((n^2 + 5*n + (3*n^2 + 13*n + 20)*x)*v[i - 1] + ((2*n + 5)*x - (2*n + 5)*x^2)*deriv(v[i - 1], x))/(n^2 + 5*n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A062190","target":"triangle","first_row":0,"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: 1,   6;\n  2: 1,  14,   21;\n  3: 1,  24,   84,    56;\n  4: 1,  36,  216,   336,    126;\n  5: 1,  50,  450,  1200,   1050,    252;\n  6: 1,  66,  825,  3300,   4950,   2772,    462;\n  7: 1,  84, 1386,  7700,  17325,  16632,   6468,    792;\n  8: 1, 104, 2184, 16016,  50050,  72072,  48048,  13728,  1287;\n  9: 1, 126, 3276, 30576, 126126, 252252, 252252, 123552, 27027, 2002;\n","recurrence_sentence":"The row polynomials satisfy (n^2+5*n)*P(n,x) = (n^2+5*n+(3*n^2+13*n+20)*x)*P(n-1,x) + ((2*n+5)*x-(2*n+5)*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[((n^2 + 5*n + (3*n^2 + 13*n + 20)*x)*P[n - 1, x] + ((2*n + 5)*x - (2*n + 5)*x^2)*D[P[n - 1, x], x])/(n^2 + 5*n)]];\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] = ((n^2 + 5*n + (3*n^2 + 13*n + 20)*x)*v[i - 1] + ((2*n + 5)*x - (2*n + 5)*x^2)*deriv(v[i - 1], x))/(n^2 + 5*n)); 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":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-5939aa09e8ea2fc4","steps":[],"edges":[],"text":"A062190"},"headline":"A062190: Coefficient triangle of certain polynomials N(5; m,x).","chain":null}
