{"id":"n-63f3fd588595ef25","revision":0,"first_index":1,"fingerprint":"f6-34367c82c455b0b6","name":"shift_x_plus_one of Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.","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"],["1","1"],["2","3","1"],["7","14","8","1"],["37","89","69","18","1"],["264","737","716","277","35","1"],["2433","7696","8892","4486","917","61","1"],["27913","98093","130450","80648","22850","2569","98","1"],["386906","1490687","2234263","1628110","587643","96154","6286","148","1"],["6346119","26542518","44135368","36892447","16133551","3508739","342260","13806","213","1"],["121159373","545598045","994208675","934974046","479323869","130529777","17440633","1058192","27804","295","1"],["2655174768","12787451789","25293138538","26375613007","15480538774","5068623913","873685694","73707953","2910677","52184","396","1"],["66028903633","338242587616","720653804872","823876832540","543843419070","208023750342","44265416226","4912121832","270921750","7265115","92411","518","1"],["1845579100993","10009697903337","22825785128916","28347512871072","20758246208900","9080287515822","2308018855416","322905206106","23641915167","884276250","16718416","155883","663","1"],["57506847262162","328904762907595","798450271120797","1069137558005276","859091202296842","422736984734180","125185306171212","21353976870756","1996428615651","99346213837","2608932755","35919286","252343","833","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","6","30","214","2030","24486","362622","6430198","133915022","3224320710","88610897694","2750125115926","95522575856942","3684465329303526"],"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":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","2","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178380,"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-b4772655514fd591"]},{"id":178381,"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-b4772655514fd591"]},{"id":178382,"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-b4772655514fd591"]}],"region_exact":{},"incoming_edges":[{"id":"e-b4772655514fd591","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-63f3fd588595ef25","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":"ea82465b60d580db944a0e064a18d1f35808cbbf450fe999476a16c46cfb40de","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-0164479e47952770","role":"sequence","name":"Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 0 <= k <= n.","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:       1,        1;\n   3:       2,        3,        1;\n   4:       7,       14,        8,        1;\n   5:      37,       89,       69,       18,        1;\n   6:     264,      737,      716,      277,       35,       1;\n   7:    2433,     7696,     8892,     4486,      917,      61,      1;\n   8:   27913,    98093,   130450,    80648,    22850,    2569,     98,     1;\n   9:  386906,  1490687,  2234263,  1628110,   587643,   96154,   6286,   148,   1;\n  10: 6346119, 26542518, 44135368, 36892447, 16133551, 3508739, 342260, 13806, 213, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) + (1+2*x+x^2)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x] + (1 + 2*x + x^2)*D[P[n - 1, x], {x, 2}]];\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] = (1 + x)*v[i - 1] + (1 + x)*deriv(v[i - 1], x) + (1 + 2*x + x^2)*deriv(deriv(v[i - 1], x), x)); 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-0164479e47952770","steps":["x \u21a6 x+1"],"edges":["e-b4772655514fd591"],"text":"A269945 \u2192 x \u21a6 x+1"},"headline":"A269945 \u2192 x \u21a6 x+1","chain":null}