{"id":"n-c73008ca3f0e3c9c","revision":0,"first_index":0,"fingerprint":"f6-c880512e8efe320e","name":"shift_x_plus_one of basis_map_A007318 of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","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"],["6","2"],["78","48","6"],["1260","1140","300","20"],["22470","26880","10920","1680","70"],["424116","631260","347760","85680","8820","252"],["8305836","14791392","10297980","3511200","595980","44352","924"],["166929048","346113768","291387096","127207080","30390360","3819816","216216","3432"],["3419932230","8091832320","7992784800","4258013760","1317115800","236396160","23063040","1029600","12870"],["71109813060","189066503340","214358189760","134888994240","51324633360","12043951920","1698977280","133024320","4813380","48620"],["1496053026468","4415675790240","5651513293140","4103268174720","1855279105680","539333803008","100333593360","11484432960","739947780","22170720","184756"],["31777397077608","103096398741336","147020024871720","120996842967000","63417814780320","22025621505888","5108306819616","777061565280","73911637800","3996272280","100876776","705432"],["680354749147164","2406504313833984","3783654718251408","3481216102684320","2076004533542940","839354619527424","233712804509184","44577167963328","5674144055580","456894197760","21059966928","454298208","2704156"],["14664155597771400","56162940057096600","96516911722140000","98180013091960800","65656477043157000","30307170851357400","9862061258649600","2268408302160000","363940322545800","39465128703000","2730656728800","108707071200","2028117000","10400600"],["317877850826299800","1310538578461963200","2443890763363303800","2723634969142651200","2018919605818915800","1048034405110492800","390590608614406200","105368562541756800","20470167383866200","2811291415732800","263468217612600","15858252446400","551242200600","8986118400","40116600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","132","2720","62020","1497888","37547664","966066816","25340180580","674628949280","18173703522832","494297477821056","13551531564682384","374024384777859200","10382416689417883200"],"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":[["0"],["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","4"],["-30","-14"],["28","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["8","12","4"],["-16","-24","-8"]]},"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-f042c28306dcf91e","recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","2"]],"recurrence":{"denominator":{"coeffs":[["0"],["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-12","-24","-12"],["32","64","32"],["-16","-32","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","2"],["-24","-8"],["24","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189458,"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-8429eaba0caeabae"]},{"id":189459,"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-8429eaba0caeabae","e-e393ec71cd9e88de"]},{"id":189460,"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-8429eaba0caeabae","e-e393ec71cd9e88de"]},{"id":189461,"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-8429eaba0caeabae","e-e393ec71cd9e88de"]},{"id":189462,"position":4,"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-e393ec71cd9e88de"]}],"region_exact":{},"incoming_edges":[{"id":"e-e393ec71cd9e88de","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-c73008ca3f0e3c9c","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":"ead236327f462de9fa0f586e5daeb55944a05b608a1deea44a7880c08fd27d91","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-ed80a7592b428c53","role":"sequence","name":"basis_map_A007318 of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","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-f042c28306dcf91e","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-c73008ca3f0e3c9c","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":"2f1a2cb9905952c0a0428ab878d2a3c377c8983c62b9245130bcf1221c840e31","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-8429eaba0caeabae","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-c73008ca3f0e3c9c","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":"388f0334347a7c6d2c297077f86353636b13bb7b9b8d75099ab7d41764d75803","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-ed80a7592b428c53","role":"sequence","name":"basis_map_A007318 of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","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-02152efcccc516c6","kind":"fingerprint_related","node_a_id":"n-939b76c3ba274588","node_b_id":"n-c73008ca3f0e3c9c","fingerprint":"f6-c880512e8efe320e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c880512e8efe320e"},"other_id":"n-939b76c3ba274588","other":{"id":"n-939b76c3ba274588","name":"mobius_x_over_x_plus_one of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","first_index":1,"fingerprint":"f6-c880512e8efe320e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","8","13"]],"card_rows":[["1"],["1","3"],["1","8","13"],["1","15","57","63"]],"row_sums_preview":["1","4","22","136","886","5944","40636","281488"],"regions":["[-1, 0)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4689fb1386d440ad","steps":["x \u21a6 x/(x+1)"],"edges":["e-5184591996606eba"],"text":"A063007 \u2192 x \u21a6 x/(x+1)"},"headline":"A063007 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-1c86f30da878a74a","kind":"fingerprint_related","node_a_id":"n-d6f28c09f7011372","node_b_id":"n-c73008ca3f0e3c9c","fingerprint":"f6-c880512e8efe320e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c880512e8efe320e"},"other_id":"n-d6f28c09f7011372","other":{"id":"n-d6f28c09f7011372","name":"mobius_x_over_x_plus_one of Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.","first_index":0,"fingerprint":"f6-c880512e8efe320e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1","3"],["0","0","1","8","13"]],"card_rows":[["1"],["0","1","3"],["0","0","1","8","13"],["0","0","0","1","15","57","63"]],"row_sums_preview":["1","4","22","136","886","5944","40636","281488"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e4d248920a317259","steps":["x \u21a6 x/(x+1)"],"edges":["e-48cb9c8ffb85d87e"],"text":"A109983 \u2192 x \u21a6 x/(x+1)"},"headline":"A109983 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-3280428eb4b6d676","kind":"fingerprint_related","node_a_id":"n-c8c2604dc29a0657","node_b_id":"n-c73008ca3f0e3c9c","fingerprint":"f6-c880512e8efe320e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c880512e8efe320e"},"other_id":"n-c8c2604dc29a0657","other":{"id":"n-c8c2604dc29a0657","name":"shift_x_plus_one of basis_map_A007318 of Square the entries of Pascal's triangle.","first_index":0,"fingerprint":"f6-c880512e8efe320e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["13","8","1"]],"card_rows":[["1"],["3","1"],["13","8","1"],["63","57","15","1"]],"row_sums_preview":["1","4","22","136","886","5944","40636","281488"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b13531ad79e7c629","steps":["x \u21a6 x+1","basis A007318"],"edges":["e-041c07f9bf4043d7","e-9ff4db41417b157a"],"text":"A008459 \u2192 x \u21a6 x+1 \u2192 basis A007318"},"headline":"A008459 \u2192 x \u21a6 x+1 \u2192 basis A007318","chain":null}},{"id":"l-43d8a7b123e33a58","kind":"fingerprint_related","node_a_id":"n-118f91023ad73cd8","node_b_id":"n-c73008ca3f0e3c9c","fingerprint":"f6-c880512e8efe320e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c880512e8efe320e"},"other_id":"n-118f91023ad73cd8","other":{"id":"n-118f91023ad73cd8","name":"basis_map_A007318 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.","first_index":1,"fingerprint":"f6-c880512e8efe320e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["13","8","1"]],"card_rows":[["1"],["3","1"],["13","8","1"],["63","57","15","1"]],"row_sums_preview":["1","4","22","136","886","5944","40636","281488"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-78ba05b4a3dd925d","steps":["basis A007318"],"edges":["e-32321abfdad5764e"],"text":"A104684 \u2192 basis A007318"},"headline":"A104684 \u2192 basis A007318","chain":null}},{"id":"l-c0f34ac28473657e","kind":"fingerprint_related","node_a_id":"n-7954740262dbc1d9","node_b_id":"n-c73008ca3f0e3c9c","fingerprint":"f6-c880512e8efe320e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c880512e8efe320e"},"other_id":"n-7954740262dbc1d9","other":{"id":"n-7954740262dbc1d9","name":"reversal of basis_map_A007318 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.","first_index":1,"fingerprint":"f6-c880512e8efe320e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","8","13"]],"card_rows":[["1"],["1","3"],["1","8","13"],["1","15","57","63"]],"row_sums_preview":["1","4","22","136","886","5944","40636","281488"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-78ba05b4a3dd925d","steps":["basis A007318","reverse"],"edges":["e-32321abfdad5764e","e-fb45aeb07dfec5df"],"text":"A104684 \u2192 basis A007318 \u2192 reverse"},"headline":"A104684 \u2192 basis A007318 \u2192 reverse","chain":null}}],"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:           6,            2;\n  2:          78,           48,            6;\n  3:        1260,         1140,          300,           20;\n  4:       22470,        26880,        10920,         1680,          70;\n  5:      424116,       631260,       347760,        85680,        8820,         252;\n  6:     8305836,     14791392,     10297980,      3511200,      595980,       44352,        924;\n  7:   166929048,    346113768,    291387096,    127207080,    30390360,     3819816,     216216,      3432;\n  8:  3419932230,   8091832320,   7992784800,   4258013760,  1317115800,   236396160,   23063040,   1029600,   12870;\n  9: 71109813060, 189066503340, 214358189760, 134888994240, 51324633360, 12043951920, 1698977280, 133024320, 4813380, 48620;\n","recurrence_sentence":"The row polynomials satisfy n^2*P(n,x) = (28*n^2-30*n+8+(12*n^2-14*n+4)*x)*P(n-1,x) - (16*n-8+(24*n-12)*x+(8*n-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[((28*n^2 - 30*n + 8 + (12*n^2 - 14*n + 4)*x)*P[n - 1, x] - (16*n - 8 + (24*n - 12)*x + (8*n - 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] = ((28*n^2 - 30*n + 8 + (12*n^2 - 14*n + 4)*x)*v[i - 1] - (16*n - 8 + (24*n - 12)*x + (8*n - 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-fdc03fad975441d0","steps":["basis A007318","x \u21a6 x+1"],"edges":["e-2cd9cd27e716475b","e-8429eaba0caeabae"],"text":"A069466 \u2192 basis A007318 \u2192 x \u21a6 x+1"},"headline":"A069466 \u2192 basis A007318 \u2192 x \u21a6 x+1","chain":null}
