{"id":"n-3c0665ea87e1dc8e","revision":2,"first_index":0,"fingerprint":"f6-b42d277a4b716fed","name":"basis_map_A007318 of A Riordan array of coordination sequences.","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"],["3","1"],["7","6","1"],["17","23","9","1"],["41","76","48","12","1"],["99","233","204","82","15","1"],["239","682","765","428","125","18","1"],["577","1935","2649","1907","775","177","21","1"],["1393","5368","8680","7656","4010","1272","238","24","1"],["3363","14641","27312","28548","18358","7506","1946","308","27","1"],["8119","39406","83313","100744","76930","38652","12908","2824","387","30","1"],["19601","104935","247985","340661","301510","180098","73920","20810","3933","475","33","1"],["47321","276996","723624","1113364","1121355","777288","379498","131272","31887","5300","572","36","1"],["114243","725849","2077164","3538998","3998245","3157539","1790302","736772","219789","46895","6952","678","39","1"],["275807","1890258","5880797","10992148","13770207","12211966","7894929","3774616","1339509","350766","66671","8916","793","42","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","14","50","178","634","2258","8042","28642","102010","363314","1293962","4608514","16413466","58457426"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":"e-1f6958db57a4f791","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["13","4"],["-9","-2"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["28","31","5"],["-8","-8","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["8","16","9","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161487,"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-69ce91caff12536e"]},{"id":161488,"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-69ce91caff12536e","e-73ab557b7db9dc16"]},{"id":161489,"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-69ce91caff12536e","e-73ab557b7db9dc16"]},{"id":161490,"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-69ce91caff12536e","e-73ab557b7db9dc16"]},{"id":161491,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-73ab557b7db9dc16"]}],"region_exact":{},"incoming_edges":[{"id":"e-69ce91caff12536e","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-3c0665ea87e1dc8e","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":"c51108c236eb1fa4290086d8a6e0dee587f8c278fe6f1f9bbe411b247ef401ec","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-ec35d1145e9a1211","role":"sequence","name":"A Riordan array of coordination sequences.","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]"}]},{"id":"e-1f6958db57a4f791","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-3c0665ea87e1dc8e","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"f3a79e75d1cb3a0a08874f1750ad17435e144ef84eebb03d4de4d082642ee218","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-73ab557b7db9dc16","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-3c0665ea87e1dc8e","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"7e6fb66ab2fd8a798ba1f9252bda02049a8a04aca760e2841b276648fbfe450d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ec35d1145e9a1211","role":"sequence","name":"A Riordan array of coordination sequences.","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-9ce5083f7519a152","kind":"fingerprint_related","node_a_id":"n-3c0665ea87e1dc8e","node_b_id":"n-7a1706417cb89af7","fingerprint":"f6-b42d277a4b716fed","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b42d277a4b716fed"},"other_id":"n-7a1706417cb89af7","other":{"id":"n-7a1706417cb89af7","name":"mobius_x_over_x_plus_one of Table a(d,m) of number of points of L1 norm m in cubic lattice Z^d, read by antidiagonals (d >= 1, m >= 0).","first_index":1,"fingerprint":"f6-b42d277a4b716fed","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","6","7"]],"card_rows":[["1"],["1","3"],["1","6","7"],["1","9","23","17"]],"row_sums_preview":["1","4","14","50","178","634","2258","8042"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e073e5e0b021f36a","steps":["x \u21a6 x/(x+1)"],"edges":["e-4b3874b4d357a754"],"text":"A035607 \u2192 x \u21a6 x/(x+1)"},"headline":"A035607 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-39e1d4d47f042b54","kind":"oeis_match","node_a_id":"n-3c0665ea87e1dc8e","node_b_id":null,"fingerprint":null,"oeis_id":"A054458","in_catalog":0,"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":false,"oeis_id":"A054458","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-5ca991695f938121","kind":"oeis_match","node_a_id":"n-3c0665ea87e1dc8e","node_b_id":null,"fingerprint":null,"oeis_id":"A055099","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":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A055099","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fb3f1f4bf213a832","kind":"oeis_match","node_a_id":"n-3c0665ea87e1dc8e","node_b_id":null,"fingerprint":null,"oeis_id":"A209696","in_catalog":0,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":false,"oeis_id":"A209696","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A209696","relation":"match_reversed_triangle"},{"oeis_id":"A055099","relation":"match_row_sums"},{"oeis_id":"A054458","relation":"match_triangle"}],"bfiles":[{"node_id":"n-3c0665ea87e1dc8e","oeis_id":"A054458","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":16521},{"node_id":"n-3c0665ea87e1dc8e","oeis_id":"A055099","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":27,"oeis_terms_available":27,"term_count":200,"raw_size":11977},{"node_id":"n-3c0665ea87e1dc8e","oeis_id":"A209696","target":"reversed_triangle","oeis_offset":1,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":16523}],"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:    3,     1;\n  2:    7,     6,     1;\n  3:   17,    23,     9,     1;\n  4:   41,    76,    48,    12,     1;\n  5:   99,   233,   204,    82,    15,    1;\n  6:  239,   682,   765,   428,   125,   18,    1;\n  7:  577,  1935,  2649,  1907,   775,  177,   21,   1;\n  8: 1393,  5368,  8680,  7656,  4010, 1272,  238,  24,  1;\n  9: 3363, 14641, 27312, 28548, 18358, 7506, 1946, 308, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) + (2+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 2, x] + (2 + 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] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 2] + (2 + 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}},{"oeis_id":"A054458","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:    3,     1;\n  2:    7,     6,     1;\n  3:   17,    23,     9,     1;\n  4:   41,    76,    48,    12,     1;\n  5:   99,   233,   204,    82,    15,    1;\n  6:  239,   682,   765,   428,   125,   18,    1;\n  7:  577,  1935,  2649,  1907,   775,  177,   21,   1;\n  8: 1393,  5368,  8680,  7656,  4010, 1272,  238,  24,  1;\n  9: 3363, 14641, 27312, 28548, 18358, 7506, 1946, 308, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) + (2+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 2, x] + (2 + x)*P[n - 1, x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 2] + (2 + x)*v[i - 1]); 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}},{"oeis_id":"A209696","target":"reversed_triangle","first_row":0,"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,  3;\n   3: 1,  6,   7;\n   4: 1,  9,  23,   17;\n   5: 1, 12,  48,   76,   41;\n   6: 1, 15,  82,  204,  233,    99;\n   7: 1, 18, 125,  428,  765,   682,   239;\n   8: 1, 21, 177,  775, 1907,  2649,  1935,   577;\n   9: 1, 24, 238, 1272, 4010,  7656,  8680,  5368,  1393;\n  10: 1, 27, 308, 1946, 7506, 18358, 28548, 27312, 14641, 3363;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+x)*P(n-2,x) + (2+x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 2, x] + (2 + x)*P[n - 1, x]];\nTable[Reverse[PadRight[CoefficientList[P[n, x], x], n]], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = (1 + x)*v[i - 2] + (2 + x)*v[i - 1]); 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":0,"offset":1,"reversed":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-ec35d1145e9a1211","steps":["basis A007318"],"edges":["e-69ce91caff12536e"],"text":"A113413 \u2192 basis A007318"},"headline":"A113413 \u2192 basis A007318","chain":null}
