{"id":"n-63fcd80b98d85584","revision":0,"first_index":0,"fingerprint":"f6-f2b1ecb4ea267ced","name":"shift_x_plus_one of A decomposition of the double factorials A001147.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["3","2"],["15","18","4"],["105","180","84","8"],["945","2100","1500","360","16"],["10395","28350","27300","10800","1488","32"],["135135","436590","529200","294000","72240","6048","64"],["2027025","7567560","11060280","7938000","2857680","463680","24384","128"],["34459425","145945800","249729480","220041360","105099120","26107200","2904000","97920","256"],["654729075","3101348250","6081075000","6356750400","3796586640","1285956000","229185600","17913600","392448","512"],["13749310575","72020198250","159202543500","192648456000","138350131920","59704555680","14922230400","1958880000","109443840","1571328","1024"],["316234143225","1814908995900","4465252291500","6140669535000","5157627274800","2713061070720","880599202560","166852224000","16437146880","664519680","6288384","2048"],["7905853580625","49332526343100","133698296031300","205977766995000","198211715301600","123217952457600","49430644623360","12402564894720","1816456561920","136148812800","4018560000","25159680","4096"],["213458046676875","1438865351673750","4259041440954300","7266895619583600","7885577930630400","5662516067208000","2712946908672000","853262368358400","168811800917760","19389904934400","1117320422400","24237158400","100651008","8192"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","5","37","377","4921","78365","1473277","31938737","784384561","21523937525","652667322517","21672312694697","782133969325801","30481907097849485"],"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":{"terms":[{"coeff":{"coeffs":[["-1","-2"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180719,"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-8e4f36558ad0ec96"]},{"id":180720,"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-8e4f36558ad0ec96"]},{"id":180721,"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-8e4f36558ad0ec96"]},{"id":180722,"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-8e4f36558ad0ec96"]}],"region_exact":{},"incoming_edges":[{"id":"e-8e4f36558ad0ec96","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-63fcd80b98d85584","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":"8b98acef7b967602fca412631a7da00c6ea79a51019bfda097fd7519f7295638","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-54b58a0ddb0e34fe","role":"sequence","name":"A decomposition of the double factorials A001147.","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-2b1b2c886b999137","kind":"fingerprint_related","node_a_id":"n-63fcd80b98d85584","node_b_id":"n-dba95db7d17b96cd","fingerprint":"f6-f2b1ecb4ea267ced","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f2b1ecb4ea267ced"},"other_id":"n-dba95db7d17b96cd","other":{"id":"n-dba95db7d17b96cd","name":"mobius_x_over_x_plus_one of A triangular decomposition of the double factorial numbers A001147.","first_index":0,"fingerprint":"f6-f2b1ecb4ea267ced","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","2","3"]],"card_rows":[["1"],["0","1"],["0","2","3"],["0","4","18","15"]],"row_sums_preview":["1","1","5","37","377","4921","78365","1473277"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-54b58a0ddb0e34fe","steps":["reverse","x \u21a6 x/(x+1)"],"edges":["e-5c085bd25a6bcbdf","e-42bbcc5b7f05b684"],"text":"A185410 \u2192 reverse \u2192 x \u21a6 x/(x+1)"},"headline":"A185410 \u2192 reverse \u2192 x \u21a6 x/(x+1)","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:        1;\n  2:        3,         2;\n  3:       15,        18,         4;\n  4:      105,       180,        84,         8;\n  5:      945,      2100,      1500,       360,        16;\n  6:    10395,     28350,     27300,     10800,      1488,       32;\n  7:   135135,    436590,    529200,    294000,     72240,     6048,      64;\n  8:  2027025,   7567560,  11060280,   7938000,   2857680,   463680,   24384,   128;\n  9: 34459425, 145945800, 249729480, 220041360, 105099120, 26107200, 2904000, 97920, 256;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+(2*n-2)*x)*P(n-1,x) - (2*x+2*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[(2*n - 1 + (2*n - 2)*x)*P[n - 1, x] - (2*x + 2*x^2)*D[P[n - 1, x], x]];\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] = (2*n - 1 + (2*n - 2)*x)*v[i - 1] - (2*x + 2*x^2)*deriv(v[i - 1], x)); 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-54b58a0ddb0e34fe","steps":["x \u21a6 x+1"],"edges":["e-8e4f36558ad0ec96"],"text":"A185410 \u2192 x \u21a6 x+1"},"headline":"A185410 \u2192 x \u21a6 x+1","chain":null}