{"id":"n-83e32d0bc1c10d45","revision":2,"first_index":0,"fingerprint":"f6-cb8a225300b7095e","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","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"],["11","7","1"],["45","42","12","1"],["197","242","104","18","1"],["903","1375","790","210","25","1"],["4279","7773","5595","2015","375","33","1"],["20793","43876","37968","17290","4410","616","42","1"],["103049","247684","250432","138040","44884","8680","952","52","1"],["518859","1399293","1618956","1048152","415044","103068","15792","1404","63","1"],["2646723","7913955","10311645","7671030","3584700","1090068","215880","27030","1995","75","1"],["13648869","44812878","64935420","54585795","29421150","10603824","2584428","420750","44055","2750","88","1"],["71039373","254064726","405273528","379930650","232114905","96827148","28081416","5652108","773685","68970","3696","102","1"],["372693519","1442131899","2511225522","2597702250","1774473129","841447893","283357932","68113188","11575707","1355783","104390","4862","117","1"],["1968801519","8195232633","15468793431","17502911517","13221767559","7027053033","2697129435","755055015","153768615","22447425","2281279","153517","6279","133","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","19","100","562","3304","20071","124996","793774","5120632","33463102","221060008","1473830308","9904186192","67015401391"],"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":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","2","1"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["3","1"],["6","2"]]},"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-0bcf293e4d350535","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","0"],["7","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4","-6","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161730,"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-3c8d43f1dfa996d6"]},{"id":161731,"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-3c8d43f1dfa996d6","e-556b2ae7482db11d"]},{"id":161732,"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-3c8d43f1dfa996d6","e-556b2ae7482db11d"]},{"id":161733,"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-3c8d43f1dfa996d6","e-556b2ae7482db11d"]},{"id":161734,"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-556b2ae7482db11d"]}],"region_exact":{},"incoming_edges":[{"id":"e-3c8d43f1dfa996d6","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-83e32d0bc1c10d45","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":"ce4378245615875252779721d09b46530c4129ef53310a527301da90cb43706a","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-57b4b14604f4c0d7","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","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-0bcf293e4d350535","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-83e32d0bc1c10d45","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":"36d0f77002f9399bfcc8e09472d23d0a6b16ab9f976f3c50ca04d5803cf13b7b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-556b2ae7482db11d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-83e32d0bc1c10d45","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":"f66efae47a2c9c94c0db1e55596480eaa13355f5a97a446563629bf7088f5b90","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-57b4b14604f4c0d7","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","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-5223938b40d3a51e","kind":"fingerprint_related","node_a_id":"n-83e32d0bc1c10d45","node_b_id":"n-5acf616616571743","fingerprint":"f6-cb8a225300b7095e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cb8a225300b7095e"},"other_id":"n-5acf616616571743","other":{"id":"n-5acf616616571743","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","first_index":0,"fingerprint":"f6-cb8a225300b7095e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","7","11"]],"card_rows":[["1"],["1","3"],["1","7","11"],["1","12","42","45"]],"row_sums_preview":["1","4","19","100","562","3304","20071","124996"],"regions":["[-1, 0)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-66e2c3448245b0b7","steps":["x \u21a6 x/(x+1)"],"edges":["e-67bfd12562549cc2"],"text":"A033282 \u2192 x \u21a6 x/(x+1)"},"headline":"A033282 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-3b510c7b3bc3c680","kind":"oeis_match","node_a_id":"n-83e32d0bc1c10d45","node_b_id":null,"fingerprint":null,"oeis_id":"A007564","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A007564","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A007564","relation":"match_row_sums"}],"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:      3,       1;\n  2:     11,       7,       1;\n  3:     45,      42,      12,       1;\n  4:    197,     242,     104,      18,      1;\n  5:    903,    1375,     790,     210,     25,      1;\n  6:   4279,    7773,    5595,    2015,    375,     33,     1;\n  7:  20793,   43876,   37968,   17290,   4410,    616,    42,    1;\n  8: 103049,  247684,  250432,  138040,  44884,   8680,   952,   52,  1;\n  9: 518859, 1399293, 1618956, 1048152, 415044, 103068, 15792, 1404, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = -(n-1+(2*n-2)*x+(n-1)*x^2)*P(n-2,x) + (6*n+3+(2*n+1)*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[Together[(-(n - 1 + (2*n - 2)*x + (n - 1)*x^2)*P[n - 2, x] + (6*n + 3 + (2*n + 1)*x)*P[n - 1, x])/(n + 2)]];\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] = (-(n - 1 + (2*n - 2)*x + (n - 1)*x^2)*v[i - 2] + (6*n + 3 + (2*n + 1)*x)*v[i - 1])/(n + 2)); 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-57b4b14604f4c0d7","steps":["basis A007318"],"edges":["e-3c8d43f1dfa996d6"],"text":"A126216 \u2192 basis A007318"},"headline":"A126216 \u2192 basis A007318","chain":null}