{"id":"n-344301f9723c6726","revision":0,"first_index":1,"fingerprint":"f6-b86e4a1bda66e0b6","name":"shift_x_plus_one of basis_map_A013609 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","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"],["33","24","4"],["245","258","84","8"],["1921","2656","1296","256","16"],["15525","26610","17360","5280","720","32"],["127905","261720","214020","88320","18960","1920","64"],["1067925","2540370","2498580","1304520","384720","62496","4928","128"],["9004545","24416640","28069440","17740800","6672960","1505280","193536","12288","256"],["76499525","232904610","306457920","227472000","104166720","30183552","5440512","571392","29952","512"],["653808673","2208268120","3272909220","2790773760","1507275840","534449664","124535040","18493440","1624320","71680","1024"],["5614995765","20835159906","34346634740","33084528840","20595136320","8646126720","2476793088","478241280","59854080","4477440","168960","2048"],["48416454529","195784325472","355333613712","381599299840","269059879440","130584655872","44446381056","10595414016","1733402880","186081280","12029952","393216","4096"],["418895174885","1833469750034","3632822970192","4303650844064","3390412659920","1868349019680","737225229312","209494112256","42479154432","5990167040","559370240","31629312","905216","8192"],["3634723102113","17119930193688","36771744590308","47636096971008","41472572149008","25586749412224","11489550456384","3794869039104","920033097984","161385928704","19886490624","1634074624","81629184","2064384","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","61","595","6145","65527","712909","7863667","87615745","983726695","11112210781","126142119187","1437751935361","16443380994775","188609259215725"],"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":[["-21","-10"],["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-7b640b1203ef46c2","recurrence":{"first_index":1,"initial_polynomials":[["1"],["5","2"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1"]]},"terms":[{"coeff":{"coeffs":[["-18","-24","-8"],["9","12","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["15","6"],["-10","-4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183252,"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-a18ade2fe8828cc3"]},{"id":183253,"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-a18ade2fe8828cc3","e-ef23e3d8341706cc"]},{"id":183254,"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-a18ade2fe8828cc3","e-ef23e3d8341706cc"]},{"id":183255,"position":3,"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-ef23e3d8341706cc"]},{"id":183256,"position":4,"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-ef23e3d8341706cc"]}],"region_exact":{},"incoming_edges":[{"id":"e-ef23e3d8341706cc","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-344301f9723c6726","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":"b95e815c56d81de03fe23cd88f6bcaf35edbe271b62733d060faca9355d39cd5","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-60c7f085d1907563","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","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-7b640b1203ef46c2","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-344301f9723c6726","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":"9b862cae7d1fcc574cfdb3619d1c4be81b6650907a3b74abef6e321df87ab1e2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-a18ade2fe8828cc3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-344301f9723c6726","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":"175c0f2ce7dbcc198dbe14d0fb48575eec473f82af5602fde164accf24b9d2bc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-60c7f085d1907563","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.","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":[{"id":"l-33b085e8be94247d","kind":"fingerprint_related","node_a_id":"n-344301f9723c6726","node_b_id":"n-72d7b1d737e4797d","fingerprint":"f6-b86e4a1bda66e0b6","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b86e4a1bda66e0b6"},"other_id":"n-72d7b1d737e4797d","other":{"id":"n-72d7b1d737e4797d","name":"shift_x_plus_one of basis_map_A007318 of Delannoy paths counted by number of weak peaks.","first_index":0,"fingerprint":"f6-b86e4a1bda66e0b6","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["33","24","4"],["245","258","84","8"],["1921","2656","1296","256","16"]],"card_rows":[["33","24","4"],["245","258","84","8"],["1921","2656","1296","256","16"],["15525","26610","17360","5280","720","32"]],"row_sums_preview":["61","595","6145","65527","712909","7863667","87615745","983726695"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b31aa1a3fd93d174","steps":["basis A007318","basis A007318"],"edges":["e-9a0dabc7f010818b","e-1bfab1fc04f48c85"],"text":"A133214 \u2192 basis A007318 \u2192 basis A007318"},"headline":"A133214 \u2192 basis A007318 \u2192 basis A007318","chain":null}}],"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:       33,        24,         4;\n   4:      245,       258,        84,         8;\n   5:     1921,      2656,      1296,       256,        16;\n   6:    15525,     26610,     17360,      5280,       720,       32;\n   7:   127905,    261720,    214020,     88320,     18960,     1920,      64;\n   8:  1067925,   2540370,   2498580,   1304520,    384720,    62496,    4928,    128;\n   9:  9004545,  24416640,  28069440,  17740800,   6672960,  1505280,  193536,  12288,   256;\n  10: 76499525, 232904610, 306457920, 227472000, 104166720, 30183552, 5440512, 571392, 29952, 512;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (13*n-21+(6*n-10)*x)*P(n-1,x) - (12+14*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[((13*n - 21 + (6*n - 10)*x)*P[n - 1, x] - (12 + 14*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] = ((13*n - 21 + (6*n - 10)*x)*v[i - 1] - (12 + 14*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-78ba05b4a3dd925d","steps":["basis A013609","x \u21a6 x+1"],"edges":["e-2f2f885dba2c5644","e-a18ade2fe8828cc3"],"text":"A104684 \u2192 basis A013609 \u2192 x \u21a6 x+1"},"headline":"A104684 \u2192 basis A013609 \u2192 x \u21a6 x+1","chain":null}
