{"id":"n-9e2a798673fae340","revision":0,"first_index":0,"fingerprint":"f6-16b10bb8625df23c","name":"one_plus_derivative of reversal of Even row Pascal-square read by antidiagonals.","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":false,"rows":[["1"],["3","1"],["7","6","1"],["14","22","9","1"],["25","62","45","12","1"],["41","147","164","76","15","1"],["63","308","486","340","115","18","1"],["92","588","1242","1210","610","162","21","1"],["129","1044","2838","3652","2535","994","217","24","1"],["175","1749","5940","9724","8827","4725","1512","280","27","1"],["231","2794","11583","23452","26845","18648","8092","2184","351","30","1"],["298","4290","21307","52195","73320","63784","35700","12996","3030","430","33","1"],["377","6370","37323","108680","183430","194616","135660","63384","19845","4070","517","36","1"],["469","9191","62712","213928","426530","540702","457368","264936","106029","29095","5324","612","39","1"],["575","12936","101660","401336","932178","1388900","1397298","978120","483483","169004","41250","6812","715","42","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","14","46","145","444","1331","3926","11434","32960","94211","267384","754309","2116936","5914310"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["10","-2","-2"],["0","0","0"],["2","0","0"]]},"terms":[{"coeff":{"coeffs":[["12","0","-4","-1"],["16","2","-4","-1"],["6","3","0","0"],["2","1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-10","-7","-1"],["0","4","1","0","0"],["0","4","1","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172917,"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-a54ee3a98ef5bd5d"]},{"id":172918,"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-a54ee3a98ef5bd5d"]},{"id":172919,"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-a54ee3a98ef5bd5d"]}],"region_exact":{},"incoming_edges":[{"id":"e-a54ee3a98ef5bd5d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-9e2a798673fae340","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"c70a897f89ac8546694594f5e50550f72a43ac7b7959668cbe3fccd3d8002a89","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-a60ffcb7522d1b36","role":"sequence","name":"reversal of Even row Pascal-square read by antidiagonals.","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-e32029a66b84240c","kind":"fingerprint_related","node_a_id":"n-3f4e0e6c6367be6f","node_b_id":"n-9e2a798673fae340","fingerprint":"f6-16b10bb8625df23c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-16b10bb8625df23c"},"other_id":"n-3f4e0e6c6367be6f","other":{"id":"n-3f4e0e6c6367be6f","name":"one_plus_derivative of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","first_index":1,"fingerprint":"f6-16b10bb8625df23c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","1"],["7","6","1"]],"card_rows":[["1"],["3","1"],["7","6","1"],["14","22","9","1"]],"row_sums_preview":["1","4","14","46","145","444","1331","3926"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-976cdb6e4a693ec8","steps":["D+1"],"edges":["e-1c757bebeb05c516"],"text":"A078812 \u2192 D+1"},"headline":"A078812 \u2192 D+1","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:   3,    1;\n  2:   7,    6,    1;\n  3:  14,   22,    9,    1;\n  4:  25,   62,   45,   12,    1;\n  5:  41,  147,  164,   76,   15,    1;\n  6:  63,  308,  486,  340,  115,   18,    1;\n  7:  92,  588, 1242, 1210,  610,  162,   21,   1;\n  8: 129, 1044, 2838, 3652, 2535,  994,  217,  24,  1;\n  9: 175, 1749, 5940, 9724, 8827, 4725, 1512, 280, 27, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+10*n-2*n*x-2*n*x^2)*P(n,x) = (2*n^3+6*n^2+16*n+12+(n^3+3*n^2+2*n)*x-(4*n+4)*x^2-(n+1)*x^3)*P(n-1,x) + ((4*n^2+4*n+8)*x+(n^2+n-10)*x^2-7*x^3-x^4)*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[Together[((2*n^3 + 6*n^2 + 16*n + 12 + (n^3 + 3*n^2 + 2*n)*x - (4*n + 4)*x^2 - (n + 1)*x^3)*P[n - 1, x] + ((4*n^2 + 4*n + 8)*x + (n^2 + n - 10)*x^2 - 7*x^3 - x^4)*D[P[n - 1, x], x])/(2*n^3 + 10*n - 2*n*x - 2*n*x^2)]];\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] = ((2*n^3 + 6*n^2 + 16*n + 12 + (n^3 + 3*n^2 + 2*n)*x - (4*n + 4)*x^2 - (n + 1)*x^3)*v[i - 1] + ((4*n^2 + 4*n + 8)*x + (n^2 + n - 10)*x^2 - 7*x^3 - x^4)*deriv(v[i - 1], x))/(2*n^3 + 10*n - 2*n*x - 2*n*x^2)); 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-79ebafb266da3b41","steps":["reverse","D+1"],"edges":["e-1e0fa78aec840e74","e-a54ee3a98ef5bd5d"],"text":"A172431 \u2192 reverse \u2192 D+1"},"headline":"A172431 \u2192 reverse \u2192 D+1","chain":null}