{"id":"n-69ac01f25ec67ed3","revision":0,"first_index":0,"fingerprint":"f6-cfb345b8c9d5b7ff","name":"reversal of basis_map_A013610 of delannoy","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","2"],["9","15","5"],["27","72","54","12"],["81","297","360","174","29"],["243","1134","1917","1476","525","70"],["729","4131","8991","9639","5409","1521","169"],["2187","14580","38880","53784","41850","18432","4284","408"],["6561","50301","158922","270216","271026","164862","59688","11820","985"],["19683","170586","623295","1259712","1553742","1215972","605934","186120","32103","2378"],["59049","570807","2368521","5552793","8150220","7838694","5015358","2114154","563697","86115","5741"],["177147","1889568","8778618","23435892","39958677","45737460","35925120","19406952","7083693","1668168","228690","13860"],["531441","6200145","31886460","95541282","185774715","247133187","230916582","152955864","71422803","22977135","4844448","602298","33461"],["1594323","20194758","113905521","378622188","827493003","1256260914","1364187177","1074213576","614326113","252435042","72575811","13850028","1575249","80782"],["4782969","65367243","401237955","1465537131","3558233691","6076634175","7532257635","6890898015","4681189665","2353181625","862951725","224212185","39077145","4095525","195025"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","29","165","941","5365","30589","174405","994381","5669525","32325149","184303845","1050819821","5991314485","34159851709"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["9"]]},"terms":[{"coeff":{"coeffs":[["0","0","36","16"],["27","-9","-60","-24"],["0","27","24","8"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","117","132","32"],["0","-27","-90","-72","-16"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","27","81","48","8"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179929,"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-bdefd448eaeefeb7"]},{"id":179930,"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-bdefd448eaeefeb7"]},{"id":179931,"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-bdefd448eaeefeb7"]}],"region_exact":{},"incoming_edges":[{"id":"e-bdefd448eaeefeb7","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-69ac01f25ec67ed3","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":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"ac3742bf0fb9b114ff4ff38c54617e8718c5107d73bbb9252025be3b04932eb3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-3594f9d1943ddb79","role":"sequence","name":"basis_map_A013610 of delannoy","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-471877eb21ebe40a","kind":"fingerprint_related","node_a_id":"n-69ac01f25ec67ed3","node_b_id":"n-0c07b65ccdfcc59b","fingerprint":"f6-cfb345b8c9d5b7ff","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cfb345b8c9d5b7ff"},"other_id":"n-0c07b65ccdfcc59b","other":{"id":"n-0c07b65ccdfcc59b","name":"reversal of basis_map_A013610 of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","first_index":1,"fingerprint":"f6-cfb345b8c9d5b7ff","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["9","15","5"]],"card_rows":[["1"],["3","2"],["9","15","5"],["27","72","54","12"]],"row_sums_preview":["1","5","29","165","941","5365","30589","174405"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-811c78acf96cbc19","steps":["basis A013610","reverse"],"edges":["e-79afa45c031edfc6","e-02ce71157bdbd564"],"text":"A008288 \u2192 basis A013610 \u2192 reverse"},"headline":"A008288 \u2192 basis A013610 \u2192 reverse","chain":null}},{"id":"l-931bdfe21170e8ec","kind":"fingerprint_related","node_a_id":"n-f17d61ee7b0cc64e","node_b_id":"n-69ac01f25ec67ed3","fingerprint":"f6-cfb345b8c9d5b7ff","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cfb345b8c9d5b7ff"},"other_id":"n-f17d61ee7b0cc64e","other":{"id":"n-f17d61ee7b0cc64e","name":"basis_map_A013610 of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.","first_index":1,"fingerprint":"f6-cfb345b8c9d5b7ff","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["5","15","9"]],"card_rows":[["1"],["2","3"],["5","15","9"],["12","54","72","27"]],"row_sums_preview":["1","5","29","165","941","5365","30589","174405"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-811c78acf96cbc19","steps":["basis A013610"],"edges":["e-79afa45c031edfc6"],"text":"A008288 \u2192 basis A013610"},"headline":"A008288 \u2192 basis A013610","chain":null}},{"id":"l-ef8281927a1753a4","kind":"fingerprint_related","node_a_id":"n-3594f9d1943ddb79","node_b_id":"n-69ac01f25ec67ed3","fingerprint":"f6-cfb345b8c9d5b7ff","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-cfb345b8c9d5b7ff"},"other_id":"n-3594f9d1943ddb79","other":{"id":"n-3594f9d1943ddb79","name":"basis_map_A013610 of delannoy","first_index":0,"fingerprint":"f6-cfb345b8c9d5b7ff","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","3"],["5","15","9"]],"card_rows":[["1"],["2","3"],["5","15","9"],["12","54","72","27"]],"row_sums_preview":["1","5","29","165","941","5365","30589","174405"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e00a1cb428776eaa","steps":["basis A013610"],"edges":["e-470609786340cbfa"],"text":"delannoy \u2192 basis A013610"},"headline":"delannoy \u2192 basis A013610","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:     3,      2;\n  2:     9,     15,      5;\n  3:    27,     72,     54,      12;\n  4:    81,    297,    360,     174,      29;\n  5:   243,   1134,   1917,    1476,     525,      70;\n  6:   729,   4131,   8991,    9639,    5409,    1521,    169;\n  7:  2187,  14580,  38880,   53784,   41850,   18432,   4284,    408;\n  8:  6561,  50301, 158922,  270216,  271026,  164862,  59688,  11820,   985;\n  9: 19683, 170586, 623295, 1259712, 1553742, 1215972, 605934, 186120, 32103, 2378;\n","recurrence_sentence":"The row polynomials satisfy 9*n*P(n,x) = (27*n+(27*n^2-9*n)*x+(24*n^2-60*n+36)*x^2+(8*n^2-24*n+16)*x^3)*P(n-1,x) - (27*n*x+(90*n-117)*x^2+(72*n-132)*x^3+(16*n-32)*x^4)*d/dx P(n-1,x) + (27*x^2+81*x^3+48*x^4+8*x^5)*d^2/dx^2 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[((27*n + (27*n^2 - 9*n)*x + (24*n^2 - 60*n + 36)*x^2 + (8*n^2 - 24*n + 16)*x^3)*P[n - 1, x] - (27*n*x + (90*n - 117)*x^2 + (72*n - 132)*x^3 + (16*n - 32)*x^4)*D[P[n - 1, x], x] + (27*x^2 + 81*x^3 + 48*x^4 + 8*x^5)*D[P[n - 1, x], {x, 2}])/(9*n)]];\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] = ((27*n + (27*n^2 - 9*n)*x + (24*n^2 - 60*n + 36)*x^2 + (8*n^2 - 24*n + 16)*x^3)*v[i - 1] - (27*n*x + (90*n - 117)*x^2 + (72*n - 132)*x^3 + (16*n - 32)*x^4)*deriv(v[i - 1], x) + (27*x^2 + 81*x^3 + 48*x^4 + 8*x^5)*deriv(deriv(v[i - 1], x), x))/(9*n)); 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-e00a1cb428776eaa","steps":["basis A013610","reverse"],"edges":["e-470609786340cbfa","e-bdefd448eaeefeb7"],"text":"delannoy \u2192 basis A013610 \u2192 reverse"},"headline":"delannoy \u2192 basis A013610 \u2192 reverse","chain":null}