{"id":"n-c551333e3802137a","revision":0,"first_index":1,"fingerprint":"f6-69e7feb8617913a4","name":"shift_x_plus_one of basis_map_A013610 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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"],["5","3"],["30","33","9"],["175","285","153","27"],["1025","2205","1764","621","81"],["6000","16050","17055","8991","2349","243"],["35125","112350","148860","104517","40986","8505","729"],["205625","765375","1214775","1065285","557199","173745","29889","2187"],["1203750","5110875","9451350","9939915","6500331","2705805","699840","102789","6561"],["7046875","33609375","70959375","87024375","68303250","35570340","12287295","2714067","347733","19683"],["41253125","218353125","518214375","726053625","664910775","415791468","179764110","53045685","10222038","1161297","59049"],["241500000","1404712500","3701756250","5832911250","6105304125","4456451115","2314342449","854952975","220128111","37614213","3838185","177147"],["1413765625","8963587500","25969050000","45454736250","53527969125","44671857615","27087260544","12022312977","3875746725","884927997","135812700","12577437","531441"],["8276328125","56807296875","179458931250","345450656250","452029308750","424541363400","294392941155","152591138739","59111172036","16900335558","3465881055","482725575","40920957","1594323"],["48450468750","357924703125","1224452559375","2570810906250","3700502566875","3862737938700","3015040671540","1787507863677","808774948953","277887074058","71364259440","13281596325","1693171026","132328809","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","72","640","5696","50688","451072","4014080","35721216","317882368","2828828672","25173688320","224020135936","1993550594048","17740565839872"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["9","3"],["-2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["15","14","3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-24f0dfd02baa8b41","recurrence":{"first_index":1,"initial_polynomials":[["1"],["5","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["5","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180018,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-6bc6677cdbee4c5b"]},{"id":180019,"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-6bc6677cdbee4c5b","e-d557303ff87f6da3"]},{"id":180020,"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-6bc6677cdbee4c5b","e-d557303ff87f6da3"]},{"id":180021,"position":3,"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-d557303ff87f6da3"]},{"id":180022,"position":4,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-d557303ff87f6da3"]}],"region_exact":{},"incoming_edges":[{"id":"e-d557303ff87f6da3","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-c551333e3802137a","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"85ff7b32c6f210b03dbdf23e47a9f5ea038e03a0a90eb936cfea6f94c837233c","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-51fcbaaf07b9691b","role":"sequence","name":"basis_map_A013610 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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]"}]},{"id":"e-24f0dfd02baa8b41","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-c551333e3802137a","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"bba46f2c506d0251962ee846d87ab424dd1f6aece4a50663e99e1fd18e4e8e22","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-6bc6677cdbee4c5b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-c551333e3802137a","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":"eb9a01df1abc8214d104a9cffd81de0196de7aa948e4f152e38db0a9661e3204","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-51fcbaaf07b9691b","role":"sequence","name":"basis_map_A013610 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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":[],"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:       5,        3;\n   3:      30,       33,        9;\n   4:     175,      285,      153,       27;\n   5:    1025,     2205,     1764,      621,       81;\n   6:    6000,    16050,    17055,     8991,     2349,      243;\n   7:   35125,   112350,   148860,   104517,    40986,     8505,      729;\n   8:  205625,   765375,  1214775,  1065285,   557199,   173745,    29889,    2187;\n   9: 1203750,  5110875,  9451350,  9939915,  6500331,  2705805,   699840,  102789,   6561;\n  10: 7046875, 33609375, 70959375, 87024375, 68303250, 35570340, 12287295, 2714067, 347733, 19683;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(2*n-9-3*x)*P(n-1,x) + (15+14*x+3*x^2)*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[(-(2*n - 9 - 3*x)*P[n - 1, x] + (15 + 14*x + 3*x^2)*D[P[n - 1, x], x])/(n - 1)]];\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] = (-(2*n - 9 - 3*x)*v[i - 1] + (15 + 14*x + 3*x^2)*deriv(v[i - 1], x))/(n - 1)); 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-c72653d9ab0ee8be","steps":["basis A013610","x \u21a6 x+1"],"edges":["e-52226b414d15e69f","e-6bc6677cdbee4c5b"],"text":"A063967 \u2192 basis A013610 \u2192 x \u21a6 x+1"},"headline":"A063967 \u2192 basis A013610 \u2192 x \u21a6 x+1","chain":null}
