{"id":"n-28e86633924230ef","revision":0,"first_index":0,"fingerprint":"f6-fe4d40ee62c150a8","name":"reversal of basis_map_A013610 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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":true,"rows":[["1"],["6","2"],["36","30","6"],["216","324","156","24"],["1296","3024","2556","924","120"],["7776","25920","33480","20880","6264","720"],["46656","209952","382320","359640","183744","48168","5040"],["279936","1632960","3973536","5216400","3979584","1758960","415584","40320"],["1679616","12317184","38537856","67060224","70825104","46370016","18324144","3982176","362880"],["10077696","90699264","354398976","787739904","1095552864","986471136","573645024","207134496","41999040","3628800"],["60466176","655050240","3124085760","8624828160","15238456128","17969908320","14293052640","7552902240","2530473696","483762240","39916800"],["362797056","4655895552","26605117440","89237998080","194928674688","290680923456","301397215680","216834045120","105817669056","33268257792","6044647680","479001600"],["2176782336","32651735040","220157346816","881294515200","2329666023168","4278158311680","5587235299008","5219168074560","3453748600896","1575239244480","468755237376","81454429440","6227020800"],["13060694016","226385362944","1778068371456","8369969946624","26316119351808","58284274194432","93417628157568","109536302633472","93790844649216","57803915879424","24865880846976","7051299899904","1177724136960","87178291200"],["78364164096","1554222587904","14064190672896","76890845251584","283446265310208","744447435443712","1434769881861888","2058482238164352","2205789607397376","1753686165014784","1016254023273216","415296012104064","112835843320320","18188931801600","1307674368000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","72","720","7920","95040","1235520","17297280","259459200","4151347200","70572902400","1270312243200","24135932620800","482718652416000","10137091700736000"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["6","1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":186625,"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-e9880b9f45e24b86"]},{"id":186626,"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-e9880b9f45e24b86"]},{"id":186627,"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-e9880b9f45e24b86"]}],"region_exact":{},"incoming_edges":[{"id":"e-e9880b9f45e24b86","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-28e86633924230ef","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":"21bb0c8dfe68e30abe690adff71288ded995932ecabb763897398fedc1754340","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-8a9c0145328414df","role":"sequence","name":"basis_map_A013610 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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-5ec91429861a32be","kind":"fingerprint_related","node_a_id":"n-8a9c0145328414df","node_b_id":"n-28e86633924230ef","fingerprint":"f6-fe4d40ee62c150a8","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fe4d40ee62c150a8"},"other_id":"n-8a9c0145328414df","other":{"id":"n-8a9c0145328414df","name":"basis_map_A013610 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","first_index":0,"fingerprint":"f6-fe4d40ee62c150a8","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","6"],["6","30","36"]],"card_rows":[["1"],["2","6"],["6","30","36"],["24","156","324","216"]],"row_sums_preview":["1","8","72","720","7920","95040","1235520","17297280"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-60c7e36cbe887b7e","steps":["basis A013610"],"edges":["e-881cf76aa24e5971"],"text":"A137320 \u2192 basis A013610"},"headline":"A137320 \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:        6,        2;\n  2:       36,       30,         6;\n  3:      216,      324,       156,        24;\n  4:     1296,     3024,      2556,       924,        120;\n  5:     7776,    25920,     33480,     20880,       6264,       720;\n  6:    46656,   209952,    382320,    359640,     183744,     48168,      5040;\n  7:   279936,  1632960,   3973536,   5216400,    3979584,   1758960,    415584,     40320;\n  8:  1679616, 12317184,  38537856,  67060224,   70825104,  46370016,  18324144,   3982176,   362880;\n  9: 10077696, 90699264, 354398976, 787739904, 1095552864, 986471136, 573645024, 207134496, 41999040, 3628800;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (6+(n+1)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(6 + (n + 1)*x)*P[n - 1, 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] = (6 + (n + 1)*x)*v[i - 1]); 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-60c7e36cbe887b7e","steps":["basis A013610","reverse"],"edges":["e-881cf76aa24e5971","e-e9880b9f45e24b86"],"text":"A137320 \u2192 basis A013610 \u2192 reverse"},"headline":"A137320 \u2192 basis A013610 \u2192 reverse","chain":null}
