{"id":"n-6148d4afd1c19fd9","revision":0,"first_index":0,"fingerprint":"f6-14fdad6b6ae97664","name":"shift_x_plus_one of basis_map_A007318 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (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"],["5","2"],["29","22","4"],["185","204","72","8"],["1257","1812","944","208","16"],["8925","15870","10920","3600","560","32"],["65445","138330","118260","51960","12240","1440","64"],["491825","1204280","1230320","675920","213920","38528","3584","128"],["3768209","10487368","12462016","8220352","3271744","797440","114688","8704","256"],["29324405","91415322","123861744","95379648","45775296","14103936","2763264","327168","20736","512"],["231153133","797815870","1213934340","1068980400","600909120","224133504","55776000","9050880","902400","48640","1024"],["1841801065","6971966276","11770123640","11666887320","7519834080","3295353600","996323328","206257920","28343040","2421760","112640","2048"],["14810069497","61006418076","113157274032","124692486160","90665364240","45669038976","16268617728","4105575936","722684160","85550080","6352896","258048","4096"],["120029657805","534494565462","1080469041496","1310293494224","1061071909040","604165669536","247953708288","73922314752","15911850240","2422058496","250398720","16347136","585728","8192"],["979470140661","4688456899266","10258885619892","13577866981224","12119274735568","7698444321568","3578281202496","1231687746432","313982532864","58615484928","7819908096","713975808","41373696","1318912","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","55","469","4237","39907","387739","3858505","39130777","402972031","4202705311","44299426717","471189693925","5051001609115","54513542257795"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","0"],["13","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-12","-14","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-11a7be78b851c6a1","recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","2"]],"recurrence":{"denominator":{"coeffs":[["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["9","12","4"],["-9","-12","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","2"],["10","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":182350,"position":0,"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-0613c6a4c63d1807"]},{"id":182351,"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-0613c6a4c63d1807","e-3f408a3e941d0619"]},{"id":182352,"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-0613c6a4c63d1807","e-3f408a3e941d0619"]},{"id":182353,"position":3,"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-3f408a3e941d0619"]},{"id":182354,"position":4,"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-3f408a3e941d0619"]}],"region_exact":{},"incoming_edges":[{"id":"e-3f408a3e941d0619","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-6148d4afd1c19fd9","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":"683d0f6c18567274872fda62fc0e90721a2fe7bebf7aa7db70aa4760a707d3dd","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-b6cda967a092feb1","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (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-11a7be78b851c6a1","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-6148d4afd1c19fd9","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":"f8ac8d15dce8ed43fff261805f793aa724b8de3d3c56ce7632dbd71b147d2e42","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-0613c6a4c63d1807","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-6148d4afd1c19fd9","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":"d6c43c61db698def2988fc9c1895f1c2f3be7e48f47e8e9f0c6a4170d03f9755","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-b6cda967a092feb1","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (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":[],"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:        5,        2;\n  2:       29,       22,         4;\n  3:      185,      204,        72,        8;\n  4:     1257,     1812,       944,      208,       16;\n  5:     8925,    15870,     10920,     3600,      560,       32;\n  6:    65445,   138330,    118260,    51960,    12240,     1440,      64;\n  7:   491825,  1204280,   1230320,   675920,   213920,    38528,    3584,    128;\n  8:  3768209, 10487368,  12462016,  8220352,  3271744,   797440,  114688,   8704,   256;\n  9: 29324405, 91415322, 123861744, 95379648, 45775296, 14103936, 2763264, 327168, 20736, 512;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = (13*n+2+6*n*x)*P(n-1,x) - (12+14*x+4*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[((13*n + 2 + 6*n*x)*P[n - 1, x] - (12 + 14*x + 4*x^2)*D[P[n - 1, x], x])/(n + 2)]];\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] = ((13*n + 2 + 6*n*x)*v[i - 1] - (12 + 14*x + 4*x^2)*deriv(v[i - 1], x))/(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-8ce61d302cec295b","steps":["basis A007318","x \u21a6 x+1"],"edges":["e-6804282af6df94ec","e-0613c6a4c63d1807"],"text":"A114687 \u2192 basis A007318 \u2192 x \u21a6 x+1"},"headline":"A114687 \u2192 basis A007318 \u2192 x \u21a6 x+1","chain":null}