{"id":"n-304dfb3caed26af2","revision":0,"first_index":1,"fingerprint":"f6-b3d4d0fb6f0d839f","name":"reversal of basis_map_A013611 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["4","1"],["16","12","2"],["64","80","28","3"],["256","448","256","60","5"],["1024","2304","1856","688","120","8"],["4096","11264","11776","6080","1664","232","13"],["16384","53248","68608","45824","17344","3760","436","21"],["65536","245760","376832","310272","151296","45120","8096","804","34"],["262144","1114112","1982464","1945600","1167360","446976","109888","16816","1460","55"],["1048576","4980736","10092544","11517952","8232960","3870720","1218304","254656","33968","2620","89"],["4194304","22020096","50069504","65208320","54214656","30330880","11699200","3123456","567680","67104","4656","144"],["16777216","96468992","243269632","356253696","338231296","219987968","100843520","32936960","7631104","1226624","130176","8208","233"],["67108864","419430400","1161822208","1890582528","2020081664","1500250112","798900224","309719040","87654400","17931776","2583424","248736","14372","377"],["268435456","1811939328","5469372416","9793699840","11639193600","9730260992","5914034176","2661138432","892928000","222842880","40803072","5325696","469232","25020","610"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","30","175","1025","6000","35125","205625","1203750","7046875","41253125","241500000","1413765625","8276328125","48450468750"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-4"],["4"]]},"terms":[{"coeff":{"coeffs":[["-16","-28","-10"],["16","16","5"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-16","-24","-5"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180391,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-0822e9fd9262324d"]},{"id":180392,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-0822e9fd9262324d"]},{"id":180393,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-0822e9fd9262324d"]}],"region_exact":{},"incoming_edges":[{"id":"e-0822e9fd9262324d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":1,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-304dfb3caed26af2","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":"fcc543ec7a5e7eb975d689bb7168199187726fbfa2e0ac7e6043942ba64d34c6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e2ca539e0aa62c2f","role":"sequence","name":"basis_map_A013611 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-45becf4ae839fdbb","kind":"fingerprint_related","node_a_id":"n-e2ca539e0aa62c2f","node_b_id":"n-304dfb3caed26af2","fingerprint":"f6-b3d4d0fb6f0d839f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b3d4d0fb6f0d839f"},"other_id":"n-e2ca539e0aa62c2f","other":{"id":"n-e2ca539e0aa62c2f","name":"basis_map_A013611 of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-b3d4d0fb6f0d839f","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","4"],["2","12","16"]],"card_rows":[["1"],["1","4"],["2","12","16"],["3","28","80","64"]],"row_sums_preview":["1","5","30","175","1025","6000","35125","205625"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-392f49b496f79fbf","steps":["basis A013611"],"edges":["e-4d9d32144b57b963"],"text":"A026729 \u2192 basis A013611"},"headline":"A026729 \u2192 basis A013611","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      4,       1;\n   3:     16,      12,       2;\n   4:     64,      80,      28,       3;\n   5:    256,     448,     256,      60,       5;\n   6:   1024,    2304,    1856,     688,     120,      8;\n   7:   4096,   11264,   11776,    6080,    1664,    232,     13;\n   8:  16384,   53248,   68608,   45824,   17344,   3760,    436,    21;\n   9:  65536,  245760,  376832,  310272,  151296,  45120,   8096,   804,   34;\n  10: 262144, 1114112, 1982464, 1945600, 1167360, 446976, 109888, 16816, 1460, 55;\n","recurrence_sentence":"The row polynomials satisfy (4*n-4)*P(n,x) = (16*n-16+(16*n-28)*x+(5*n-10)*x^2)*P(n-1,x) - (16*x+24*x^2+5*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n - 16 + (16*n - 28)*x + (5*n - 10)*x^2)*P[n - 1, x] - (16*x + 24*x^2 + 5*x^3)*D[P[n - 1, x], x])/(4*n - 4)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((16*n - 16 + (16*n - 28)*x + (5*n - 10)*x^2)*v[i - 1] - (16*x + 24*x^2 + 5*x^3)*deriv(v[i - 1], x))/(4*n - 4)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-392f49b496f79fbf","steps":["basis A013611","reverse"],"edges":["e-4d9d32144b57b963","e-0822e9fd9262324d"],"text":"A026729 \u2192 basis A013611 \u2192 reverse"},"headline":"A026729 \u2192 basis A013611 \u2192 reverse","chain":null}