{"id":"n-3059d210b3bacecf","revision":0,"first_index":1,"fingerprint":"f6-1aca77fc72184605","name":"shift_x_plus_one of basis_map_A013609 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 peaks (1 <= k <= n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["5","2"],["31","24","4"],["215","246","84","8"],["1597","2416","1256","256","16"],["12425","23370","16360","5200","720","32"],["99955","224760","198060","86080","18960","1920","64"],["824675","2157470","2293340","1268120","386960","63392","4928","128"],["6939769","20705184","25780944","17332224","6791904","1559040","198912","12288","256"],["59334605","198823122","283837392","224878752","107890272","32066496","5789952","594432","29952","512"],["513972967","1911025240","3077187540","2808662400","1596682080","584656128","137692800","20183040","1708800","71680","1024"],["4501041935","18388387974","32970695780","34076276520","22414566240","9773184960","2851744896","548824320","66865920","4759040","168960","2048"],["39784038517","177140321808","350014813368","404120907520","302078121840","153007861248","53402418432","12798922752","2059580160","212439040","12908544","393216","4096"],["354455513105","1708371390362","3688215746488","4705531406576","3941056399280","2275962745056","926167866624","266600176128","53677029120","7352423936","651622400","34238464","905216","8192"],["3179928556219","16493792323512","38627910285692","53972058026432","50076995400432","32492776091776","15121342332096","5092190078976","1236045947136","212747831296","25163314176","1939718144","89083904","2064384","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","59","553","5541","58107","629799","6999013","79320521","913245487","10651843699","125596518593","1494632730541","17928077470947","216532981070559"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-7","-6"],["11","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","-8","-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-9db5e9289aa03213","recurrence":{"first_index":1,"initial_polynomials":[["1"],["5","2"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","8","8"],["-1","-4","-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":183202,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-bfbf715c9011eff4"]},{"id":183203,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-bfbf715c9011eff4","e-c969a4514b0221fe"]},{"id":183204,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-bfbf715c9011eff4","e-c969a4514b0221fe"]},{"id":183205,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-bfbf715c9011eff4","e-c969a4514b0221fe"]},{"id":183206,"position":4,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-c969a4514b0221fe"]}],"region_exact":{},"incoming_edges":[{"id":"e-c969a4514b0221fe","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-3059d210b3bacecf","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":"2849ba3244d4f6ba192551e8039de7a50f71f67f71c658674ece6d3d22a0407b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4d7e6ff489a3ffc4","role":"sequence","name":"basis_map_A013609 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 peaks (1 <= k <= n).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-9db5e9289aa03213","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-3059d210b3bacecf","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":"9dea3de0dcee2e03a322d5f51f85843acd0e7b2ab060750a261a4f8f1f9e11b8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-bfbf715c9011eff4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-3059d210b3bacecf","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":"37ac62a06faa83f39acdfc786882263af5271176bbc00c1fd09b3645f96102d8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-4d7e6ff489a3ffc4","role":"sequence","name":"basis_map_A013609 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 peaks (1 <= k <= n).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        1;\n   2:        5,         2;\n   3:       31,        24,         4;\n   4:      215,       246,        84,         8;\n   5:     1597,      2416,      1256,       256,        16;\n   6:    12425,     23370,     16360,      5200,       720,       32;\n   7:    99955,    224760,    198060,     86080,     18960,     1920,      64;\n   8:   824675,   2157470,   2293340,   1268120,    386960,    63392,    4928,    128;\n   9:  6939769,  20705184,  25780944,  17332224,   6791904,  1559040,  198912,  12288,   256;\n  10: 59334605, 198823122, 283837392, 224878752, 107890272, 32066496, 5789952, 594432, 29952, 512;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = (11*n-7+(6*n-6)*x)*P(n-1,x) - (3+8*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((11*n - 7 + (6*n - 6)*x)*P[n - 1, x] - (3 + 8*x + 4*x^2)*D[P[n - 1, x], x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((11*n - 7 + (6*n - 6)*x)*v[i - 1] - (3 + 8*x + 4*x^2)*deriv(v[i - 1], x))/(n + 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-0d75f6272d2a2a84","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-077b645e97fdde1b","e-bfbf715c9011eff4"],"text":"A114656 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A114656 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}