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Polya and I. Schur, J. Reine Angew. Math. 144 (1914) 89--113; stated as Theorem 40 of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601): a real sequence (lambda_k) is a multiplier sequence (x^k -> lambda_k x^k maps real-rooted polynomials to real-rooted polynomials or zero) iff sum_k binom(m,k) lambda_k x^k is real-rooted with zeros of one sign for every m, iff sum_k lambda_k x^k / k! (or its reflection) is C x^m e^(a x) prod (1 + alpha_j x) with a, alpha_j >= 0; a product of multiplier sequences is one (composition), and a nonzero factor lambda(n) of the row changes nothing","lean_route":null,"lean_declarations":[],"certificate_hash":"d730327b585ad78c0f10a988767d2f30b08f06bf5831c2e72dd69e0744339684","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-f7935c36a3c0ec8c","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(2n, 2k).","uses":{"claim":"real_rooted","from_index":3,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 3, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-17458235c0f86642","kind":"oeis_match","node_a_id":"n-b002679da3d71a0f","node_b_id":null,"fingerprint":null,"oeis_id":"A288470","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A288470","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A288470","relation":"match_row_sums"},{"oeis_id":"A155495","relation":"tail"}],"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,   12,      1;\n   2: 1,   45,     45,        1;\n   3: 1,  112,    420,      112,         1;\n   4: 1,  225,   2100,     2100,       225,         1;\n   5: 1,  396,   7425,    18480,      7425,       396,         1;\n   6: 1,  637,  21021,   105105,    105105,     21021,       637,         1;\n   7: 1,  960,  50960,   448448,    900900,    448448,     50960,       960,        1;\n   8: 1, 1377, 110160,  1559376,   5513508,   5513508,   1559376,    110160,     1377,      1;\n   9: 1, 1900, 218025,  4651200,  26453700,  46558512,  26453700,   4651200,   218025,   1900,    1;\n  10: 1, 2541, 402325, 12311145, 105524100, 298750452, 298750452, 105524100, 12311145, 402325, 2541, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^2+3*n+1)*P(n,x) = (30*n^2-11*n+1-(24*n^2-37*n-1)*x)*P(n-1,x) - ((54*n-48)*x-(54*n-48)*x^2)*d/dx P(n-1,x) - (36*n^2-34*n+12-(144*n^2-124*n+24)*x+(36*n^2+14*n-36)*x^2)*P(n-2,x) - (48*x-48*x^3)*d/dx P(n-2,x) + (8*n^2-20*n+12+(24*n^2-60*n+36)*x+(24*n^2-60*n+36)*x^2+(8*n^2-20*n+12)*x^3)*P(n-3,x) for n >= 4, with P(1,x) = 1 + 12*x + x^2, P(2,x) = 1 + 45*x + 45*x^2 + x^3, P(3,x) = 1 + 112*x + 420*x^2 + 112*x^3 + x^4.","mathematica":"P[1, x_] := 1 + 12*x + x^2;\nP[2, x_] := 1 + 45*x + 45*x^2 + x^3;\nP[3, x_] := 1 + 112*x + 420*x^2 + 112*x^3 + x^4;\nP[n_, x_] := P[n, x] = Expand[Together[((30*n^2 - 11*n + 1 - (24*n^2 - 37*n - 1)*x)*P[n - 1, x] - ((54*n - 48)*x - (54*n - 48)*x^2)*D[P[n - 1, x], x] - (36*n^2 - 34*n + 12 - (144*n^2 - 124*n + 24)*x + (36*n^2 + 14*n - 36)*x^2)*P[n - 2, x] - (48*x - 48*x^3)*D[P[n - 2, x], x] + (8*n^2 - 20*n + 12 + (24*n^2 - 60*n + 36)*x + (24*n^2 - 60*n + 36)*x^2 + (8*n^2 - 20*n + 12)*x^3)*P[n - 3, x])/(2*n^2 + 3*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 12*x + x^2; v[2] = 1 + 45*x + 45*x^2 + x^3; v[3] = 1 + 112*x + 420*x^2 + 112*x^3 + x^4; for(n = 4, N, my(i = n); v[i] = ((30*n^2 - 11*n + 1 - (24*n^2 - 37*n - 1)*x)*v[i - 1] - ((54*n - 48)*x - (54*n - 48)*x^2)*deriv(v[i - 1], x) - (36*n^2 - 34*n + 12 - (144*n^2 - 124*n + 24)*x + (36*n^2 + 14*n - 36)*x^2)*v[i - 2] - (48*x - 48*x^3)*deriv(v[i - 2], x) + (8*n^2 - 20*n + 12 + (24*n^2 - 60*n + 36)*x + (24*n^2 - 60*n + 36)*x^2 + (8*n^2 - 20*n + 12)*x^3)*v[i - 3])/(2*n^2 + 3*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-f7935c36a3c0ec8c","steps":["\u2299 C(n+1,k) (shift +2)"],"edges":["e-e37e9c5997239324"],"text":"A086645 \u2192 \u2299 C(n+1,k) (shift +2)"},"headline":"A086645 \u2192 \u2299 C(n+1,k) (shift +2)","chain":null}