{"id":"n-47057799697d7ab0","revision":0,"first_index":0,"fingerprint":"f6-1f4fdff1bda9ed4e","name":"shift_x_plus_one of shift_x_plus_one of Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["5","2"],["17","8"],["109","86","16"],["649","596","136"],["5285","6258","2352","272"],["44513","60144","26880","3968"],["448861","712558","407904","97792","7936"],["4836601","8586508","5667144","1645312","176896"],["58743125","117281962","90778016","33602768","5803264","353792"],["766520657","1682409064","1463753312","629942336","133794688","11184128"],["10939702669","26350633638","25782084720","13000043840","3509614080","467244288","22368256"],["167136559849","436576035588","470870441160","267975142400","84687671040","14049627648","951878656"],["2746173173765","7751067555938","9179736948816","5876959501840","2175364360960","457827095040","49078255616","1903757312"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","7","25","211","1381","14167","135505","1675051","20912461","306562927","4687604185","80071691491","1442247356341","28238110649287"],"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":[["-3","-2"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-14","-15","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178773,"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-7d132b42f43f5cfb"]},{"id":178774,"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-7d132b42f43f5cfb"]},{"id":178775,"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-7d132b42f43f5cfb"]}],"region_exact":{},"incoming_edges":[{"id":"e-7d132b42f43f5cfb","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-47057799697d7ab0","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":"33e4b7ac28548dea8665b1d45da1bf50582d663b62b26192fdd0d1331dce3706","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-e8d6574c8023ca59","role":"sequence","name":"shift_x_plus_one of Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).","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":[],"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:       5,       2;\n  3:      17,       8;\n  4:     109,      86,      16;\n  5:     649,     596,     136;\n  6:    5285,    6258,    2352,     272;\n  7:   44513,   60144,   26880,    3968;\n  8:  448861,  712558,  407904,   97792,   7936;\n  9: 4836601, 8586508, 5667144, 1645312, 176896;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n-3+(2*n-2)*x)*P(n-1,x) - (14+15*x+4*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[(4*n - 3 + (2*n - 2)*x)*P[n - 1, x] - (14 + 15*x + 4*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] = (4*n - 3 + (2*n - 2)*x)*v[i - 1] - (14 + 15*x + 4*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-ff270bcb802a85af","steps":["\u03b3 (shift +1)","x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-7e5061201b6d9b87","e-33f9ec87155c6d6e","e-7d132b42f43f5cfb"],"text":"A008292 \u2192 \u03b3 (shift +1) \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"A008292 \u2192 \u03b3 (shift +1) \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}
