{"id":"n-ff73176658f063e0","revision":0,"first_index":0,"fingerprint":"f6-8deca67d09eebe9e","name":"reversal of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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"],["1","15"],["1","30","225"],["1","45","675","3375"],["1","60","1350","13500","50625"],["1","75","2250","33750","253125","759375"],["1","90","3375","67500","759375","4556250","11390625"],["1","105","4725","118125","1771875","15946875","79734375","170859375"],["1","120","6300","189000","3543750","42525000","318937500","1366875000","2562890625"],["1","135","8100","283500","6378750","95681250","956812500","6150937500","23066015625","38443359375"],["1","150","10125","405000","10631250","191362500","2392031250","20503125000","115330078125","384433593750","576650390625"],["1","165","12375","556875","16706250","350831250","5262468750","56383593750","422876953125","2114384765625","6343154296875","8649755859375"],["1","180","14850","742500","25059375","601425000","10524937500","135320625000","1268630859375","8457539062500","38058925781250","103797070312500","129746337890625"],["1","195","17550","965250","36196875","977315625","19546312500","293194687500","3298440234375","27487001953125","164922011718750","674680957031250","1686702392578125","1946195068359375"],["1","210","20475","1228500","50675625","1520268750","34206046875","586389375000","7696360546875","76963605468750","577227041015625","3148511132812500","11806916748046875","27246730957031250","29192926025390625"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","16","256","4096","65536","1048576","16777216","268435456","4294967296","68719476736","1099511627776","17592186044416","281474976710656","4503599627370496","72057594037927936"],"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":[["1","15"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190124,"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-d8eb2b23fe746b9c"]},{"id":190125,"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-d8eb2b23fe746b9c"]},{"id":190126,"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-d8eb2b23fe746b9c"]}],"region_exact":{},"incoming_edges":[{"id":"e-d8eb2b23fe746b9c","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-ff73176658f063e0","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":"2745ba11558dc2ff412949fd6f9634fe49bda8ebf2386ece7265067f2c853544","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-ff0214c333bfdcca","role":"sequence","name":"Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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-2d5b694b5839ec50","kind":"fingerprint_related","node_a_id":"n-236c1003708dee15","node_b_id":"n-ff73176658f063e0","fingerprint":"f6-8deca67d09eebe9e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8deca67d09eebe9e"},"other_id":"n-236c1003708dee15","other":{"id":"n-236c1003708dee15","name":"derivative of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-8deca67d09eebe9e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],["30","2"]],"card_rows":[["1"],["30","2"],["675","90","3"],["13500","2700","180","4"]],"row_sums_preview":["0","1","32","768","16384","327680","6291456","117440512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ff0214c333bfdcca","steps":["D"],"edges":["e-b8062bd651f856c8"],"text":"A027467 \u2192 D"},"headline":"A027467 \u2192 D","chain":null}},{"id":"l-4a1a53fd530bae4d","kind":"fingerprint_related","node_a_id":"n-ff0214c333bfdcca","node_b_id":"n-ff73176658f063e0","fingerprint":"f6-8deca67d09eebe9e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-8deca67d09eebe9e"},"other_id":"n-ff0214c333bfdcca","other":{"id":"n-ff0214c333bfdcca","name":"Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","first_index":0,"fingerprint":"f6-8deca67d09eebe9e","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["15","1"],["225","30","1"]],"card_rows":[["1"],["15","1"],["225","30","1"],["3375","675","45","1"]],"row_sums_preview":["1","16","256","4096","65536","1048576","16777216","268435456"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A027467"],"derivation":{"root":"n-ff0214c333bfdcca","steps":[],"edges":[],"text":"A027467"},"headline":"A027467: Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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,  15;\n  2: 1,  30,  225;\n  3: 1,  45,  675,   3375;\n  4: 1,  60, 1350,  13500,   50625;\n  5: 1,  75, 2250,  33750,  253125,   759375;\n  6: 1,  90, 3375,  67500,  759375,  4556250,  11390625;\n  7: 1, 105, 4725, 118125, 1771875, 15946875,  79734375,  170859375;\n  8: 1, 120, 6300, 189000, 3543750, 42525000, 318937500, 1366875000,  2562890625;\n  9: 1, 135, 8100, 283500, 6378750, 95681250, 956812500, 6150937500, 23066015625, 38443359375;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+15*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[(1 + 15*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] = (1 + 15*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-ff0214c333bfdcca","steps":["reverse"],"edges":["e-d8eb2b23fe746b9c"],"text":"A027467 \u2192 reverse"},"headline":"A027467 \u2192 reverse","chain":null}
