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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":"2b1ddd0f9634c908ab17fb74ec4d3fa52913b0569fddefdc3ca3b603811240ef","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-5c0c5bce3aee8449","role":"sequence","name":"Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 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":[{"id":"l-e6764b4b3bd605d2","kind":"oeis_match","node_a_id":"n-a05d8afbacbf6624","node_b_id":null,"fingerprint":null,"oeis_id":"A262442","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A262442","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A262442","relation":"match_row_sums"}],"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:  2,    1;\n   3:  5,    6,     1;\n   4: 10,   29,    13,      1;\n   5: 17,  100,   102,     24,      1;\n   6: 26,  265,   540,    280,     40,      1;\n   7: 37,  586,  2075,   2112,    658,     62,     1;\n   8: 50, 1141,  6321,  11165,   6720,   1386,    91,    1;\n   9: 65, 2024, 16268,  45136,  46970,  18480,  2688,  128,   1;\n  10: 82, 3345, 36912, 149352, 244608, 165594, 45540, 4884, 174, 1;\n","recurrence_sentence":"The row polynomials satisfy (16*n^6-96*n^5+212*n^4-164*n^3-28*n^2+60*n+(48*n^5-256*n^4+380*n^3-28*n^2-144*n)*x+(48*n^4-164*n^3+46*n^2+70*n)*x^2+(16*n^3-8*n^2-8*n)*x^3)*P(n,x) = (16*n^6-64*n^5+100*n^4-60*n^3-16*n^2+24*n+(144*n^6-1192*n^5+4112*n^4-7556*n^3+7808*n^2-4132*n+816)*x+(432*n^5-3240*n^4+9240*n^3-12522*n^2+7644*n-1464)*x^2+(432*n^4-2492*n^3+4908*n^2-3706*n+744)*x^3+(144*n^3-464*n^2+440*n-96)*x^4)*P(n-1,x) + ((32*n^5-224*n^4+588*n^3-776*n^2+500*n-120)*x-(208*n^5-1520*n^4+4312*n^3-6132*n^2+4182*n-1014)*x^2-(624*n^4-3856*n^3+8220*n^2-7016*n+1746)*x^3-(624*n^3-2852*n^2+3698*n-996)*x^4-(208*n^2-472*n+144)*x^5)*d/dx P(n-1,x) - ((20*n^4-82*n^3+140*n^2-108*n+30)*x^2-(80*n^4-388*n^3+770*n^2-662*n+188)*x^3-(240*n^3-900*n^2+1059*n-318)*x^4-(240*n^2-576*n+192)*x^5-(80*n-32)*x^6)*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[Together[((16*n^6 - 64*n^5 + 100*n^4 - 60*n^3 - 16*n^2 + 24*n + (144*n^6 - 1192*n^5 + 4112*n^4 - 7556*n^3 + 7808*n^2 - 4132*n + 816)*x + (432*n^5 - 3240*n^4 + 9240*n^3 - 12522*n^2 + 7644*n - 1464)*x^2 + (432*n^4 - 2492*n^3 + 4908*n^2 - 3706*n + 744)*x^3 + (144*n^3 - 464*n^2 + 440*n - 96)*x^4)*P[n - 1, x] + ((32*n^5 - 224*n^4 + 588*n^3 - 776*n^2 + 500*n - 120)*x - (208*n^5 - 1520*n^4 + 4312*n^3 - 6132*n^2 + 4182*n - 1014)*x^2 - (624*n^4 - 3856*n^3 + 8220*n^2 - 7016*n + 1746)*x^3 - (624*n^3 - 2852*n^2 + 3698*n - 996)*x^4 - (208*n^2 - 472*n + 144)*x^5)*D[P[n - 1, x], x] - ((20*n^4 - 82*n^3 + 140*n^2 - 108*n + 30)*x^2 - (80*n^4 - 388*n^3 + 770*n^2 - 662*n + 188)*x^3 - (240*n^3 - 900*n^2 + 1059*n - 318)*x^4 - (240*n^2 - 576*n + 192)*x^5 - (80*n - 32)*x^6)*D[P[n - 1, x], {x, 2}])/(16*n^6 - 96*n^5 + 212*n^4 - 164*n^3 - 28*n^2 + 60*n + (48*n^5 - 256*n^4 + 380*n^3 - 28*n^2 - 144*n)*x + (48*n^4 - 164*n^3 + 46*n^2 + 70*n)*x^2 + (16*n^3 - 8*n^2 - 8*n)*x^3)]];\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^6 - 64*n^5 + 100*n^4 - 60*n^3 - 16*n^2 + 24*n + (144*n^6 - 1192*n^5 + 4112*n^4 - 7556*n^3 + 7808*n^2 - 4132*n + 816)*x + (432*n^5 - 3240*n^4 + 9240*n^3 - 12522*n^2 + 7644*n - 1464)*x^2 + (432*n^4 - 2492*n^3 + 4908*n^2 - 3706*n + 744)*x^3 + (144*n^3 - 464*n^2 + 440*n - 96)*x^4)*v[i - 1] + ((32*n^5 - 224*n^4 + 588*n^3 - 776*n^2 + 500*n - 120)*x - (208*n^5 - 1520*n^4 + 4312*n^3 - 6132*n^2 + 4182*n - 1014)*x^2 - (624*n^4 - 3856*n^3 + 8220*n^2 - 7016*n + 1746)*x^3 - (624*n^3 - 2852*n^2 + 3698*n - 996)*x^4 - (208*n^2 - 472*n + 144)*x^5)*deriv(v[i - 1], x) - ((20*n^4 - 82*n^3 + 140*n^2 - 108*n + 30)*x^2 - (80*n^4 - 388*n^3 + 770*n^2 - 662*n + 188)*x^3 - (240*n^3 - 900*n^2 + 1059*n - 318)*x^4 - (240*n^2 - 576*n + 192)*x^5 - (80*n - 32)*x^6)*deriv(deriv(v[i - 1], x), x))/(16*n^6 - 96*n^5 + 212*n^4 - 164*n^3 - 28*n^2 + 60*n + (48*n^5 - 256*n^4 + 380*n^3 - 28*n^2 - 144*n)*x + (48*n^4 - 164*n^3 + 46*n^2 + 70*n)*x^2 + (16*n^3 - 8*n^2 - 8*n)*x^3)); 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-79ebafb266da3b41","steps":["\u2299 C(n,k+1) (shift -1)","D+1"],"edges":["e-1491609a6fc5f8c9","e-f5f4cf0c3e186e1f"],"text":"A172431 \u2192 \u2299 C(n,k+1) (shift -1) \u2192 D+1"},"headline":"A172431 \u2192 \u2299 C(n,k+1) (shift -1) \u2192 D+1","chain":null}