{"id":"n-a30a6965ac143471","revision":0,"first_index":0,"fingerprint":"f6-450e847a2555bc30","name":"basis_map_A007318 of basis_map_A007318 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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"],["7","3"],["25","18","3"],["103","105","33","3"],["409","540","252","48","3"],["1639","2655","1638","474","63","3"],["6553","12582","9657","3756","771","78","3"],["26215","58197","53415","26097","7269","1143","93","3"],["104857","264240","282234","166044","58500","12552","1590","108","3"],["419431","1182795","1440996","991584","422814","115266","19980","2112","123","3"],["1677721","5234490","7163199","5644392","2826114","935820","206670","29928","2709","138","3"],["6710887","22951233","34853661","30944703","17797158","6940266","1866282","344862","42771","3381","153","3"],["26843545","99857988","166633992","164591736","106929417","48013464","15237432","3433920","543867","58884","4128","168","3"],["107374183","431644215","785077074","853895334","618458025","314325189","115071420","30651204","5926329","819585","78642","4950","183","3"],["429496729","1855425870","3653127381","4338398004","3466170549","1967346282","816516105","251235576","57473499","9708354","1189791","102420","5847","198","3"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","46","244","1252","6472","33400","172432","890128","4595104","23721184","122455360","632148544","3263326336","16846196608"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["7","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195431,"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":true,"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-c8cd2ecaeb3a3571"]},{"id":195432,"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-c8cd2ecaeb3a3571"]},{"id":195433,"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-c8cd2ecaeb3a3571"]},{"id":195434,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-c8cd2ecaeb3a3571"]}],"region_exact":{},"incoming_edges":[{"id":"e-c8cd2ecaeb3a3571","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-a30a6965ac143471","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"6aca989fbb22a6add2d645369fa409d00b20a7a0a1a6ce3ea356e5f40602b566","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-216193c52744946f","role":"sequence","name":"basis_map_A007318 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-5659e71a102eb869","kind":"candidate_equal","node_a_id":"n-96ebc83d17e3d946","node_b_id":"n-a30a6965ac143471","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-96ebc83d17e3d946","other":{"id":"n-96ebc83d17e3d946","name":"shift_x_plus_one of basis_map_A007318 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","first_index":0,"fingerprint":"f6-450e847a2555bc30","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["7","3"],["25","18","3"]],"card_rows":[["1"],["7","3"],["25","18","3"],["103","105","33","3"]],"row_sums_preview":["1","10","46","244","1252","6472","33400","172432"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-eac80c2bf8e566f5","steps":["x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-237035d3e801d2dc","e-3be841aa42c5b407"],"text":"A249250 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A249250 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}}],"oeis":[],"bfiles":[],"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:      7,       3;\n  2:     25,      18,       3;\n  3:    103,     105,      33,      3;\n  4:    409,     540,     252,     48,      3;\n  5:   1639,    2655,    1638,    474,     63,      3;\n  6:   6553,   12582,    9657,   3756,    771,     78,     3;\n  7:  26215,   58197,   53415,  26097,   7269,   1143,    93,    3;\n  8: 104857,  264240,  282234, 166044,  58500,  12552,  1590,  108,   3;\n  9: 419431, 1182795, 1440996, 991584, 422814, 115266, 19980, 2112, 123, 3;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+2*x)*P(n-2,x) + (3+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 7 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(4 + 2*x)*P[n - 2, x] + (3 + x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 7 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 2*x)*v[i - 2] + (3 + x)*v[i - 1]); 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}}],"derivation":{"root":"n-eac80c2bf8e566f5","steps":["x \u21a6 x+1","basis A007318"],"edges":["e-237035d3e801d2dc","e-c8cd2ecaeb3a3571"],"text":"A249250 \u2192 x \u21a6 x+1 \u2192 basis A007318"},"headline":"A249250 \u2192 x \u21a6 x+1 \u2192 basis A007318","chain":null}